INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1974

INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1974

GLOBAL ANALYSIS AND ITS APPLICATIONS

Vol. II

INTERN ATIO N AL C E N T R E FO R T H EO R ET IC A L PH YSICS, T R IE ST E

GLOBAL ANALYSIS AND ITS APPLICATIONS

L E C T U R E S P R E SE N T E D AT AN INTERN ATIO N AL SEM INAR COURSE

AT T R IE S T E FROM 4 JU L Y TO 25 AUGUST 1972 ORGANIZED BY THE

INTERN ATIO N AL C E N T R E FO R T H EO R ET IC A L PH YSICS, T R IE ST E

In three volumes

VOL. II

IN TERN ATIO N AL ATOMIC EN ERG Y AGENCY VIENNA, 1974

THE INTERN ATIO N AL C EN TR E FOR T H EO R ET IC A L PH YSICS (ICTP) in T r ie s te w as e sta b lish e d b y the International A tom ic E n ergy A gency (IAEA) in 1964 under an agreem en t with the Italian G overnm ent, and with the a s s i s tance of the City and U n iversity of T r ie s te .

The IAEA and the United N ations E ducation al, Sc ien tific and C ultural O rganization (UNESCO) subsequently ag reed to op erate the Centre jo in tly from1 Ja n u a ry 1970.

M em ber S ta te s of both o rgan izatio n s p artic ip a te in the work of the C entre, the m ain pu rpose of which is to fo s te r , through tra in in g and re se a r c h , the advancem ent of th eo re tica l p h y sic s , with sp e c ia l re g a rd to the needs of developing cou n tries.

G LO BA L ANALYSIS AND ITS A PPLICA TIO N S IAEA, VIENNA, 1974

S T I/P U B /3 5 5

Printed by the IAEA in Austria September 1974

FOREWORD

The International C entre fo r T h eo re tica l P h y sics has m aintained an in te rd isc ip lin a ry ch a ra c te r in it s r e se a r c h and train in g, p ro g ram m es in d ifferen t branches o f th eo re tica l p h y sic s. In pursuan ce of th is ob jective , the C entre h as organ ized extended r e se a r c h c o u rse s with a com preh ensive and synoptic cov erage in vary in g d isc ip lin e s . The f i r s t of these — on p la sm a p h y sics — w as held in 1964; the secon d , in 1965, w as concerned with the p h y sic s of p a r t ic le s . Between then and 1972, seven c o u rse s w ere organ ized ; three on n u clear theory (during 1966, 1969 and 1971), three on ph y sics of condensed m atte r (during 1967, 1970 and 1972), and one on Com puting a s a Language of P h y sic s (1971). The P ro ceed in gs of a ll th ese c o u rse s have been published by the International A tom ic E nergy Agency.

The p resen t three vo lum es re c o rd the P ro ceed in gs o f the tenth co u rse held from 4 Ju ly to 25 A ugust 1972 which dealt with G lobal A n aly sis and it s A pp lication s. G enerous g ran ts from the United N ations D evelopm ent P r o gram m e and from the B atte lle Foundation a re gratefu lly acknow ledged. The p rogram m e of le c tu re s w as organ ized by P r o fe s s o r s M. D olcher (T r ie s te , Italy), J . E e lls (W arwick, United Kingdom) and J . C . Zeem an (W arw ick, United Kingdom ).

Abdus Salam

EDITORIAL NOTE

T h e p a p e r s a n d d i s c u s s i o n s i n c o r p o r a t e d in t h e p r o c e e d i n g s p u b l i s h e d

b y t h e I n t e r n a t i o n a l A t o m i c E n e r g y A g e n c y a r e e d i t e d b y t h e A g e n c y ' s e d i

t o r i a l s t a f f to t h e e x t e n t c o n s i d e r e d n e c e s s a r y f o r t h e r e a d e r ' s a s s i s t a n c e .

T h e v i e w s e x p r e s s e d a n d t h e g e n e r a l s t y l e a d o p t e d r e m a i n , h o w e v e r , t h e

r e s p o n s i b i l i t y o f t h e n a m e d a u t h o r s o r p a r t i c i p a n t s .

F o r t h e s a k e o f s p e e d o f p u b l i c a t i o n t h e p r e s e n t P r o c e e d i n g s h a v e b e e n

p r i n t e d b y c o m p o s i t i o n t y p i n g a n d p h o t o - o f f s e t l i t h o g r a p h y . W ithin th e l i m i

t a t i o n s i m p o s e d b y t h i s m e t h o d , e v e r y e f f o r t h a s b e e n m a d e to m a i n t a i n a

h i g h e d i t o r i a l s t a n d a r d ; i n p a r t i c u l a r , t h e u n i t s a n d s y m b o l s e m p l o y e d a r e

to t h e f u l l e s t p r a c t i c a b l e e x t e n t t h o s e s t a n d a r d i z e d o r r e c o m m e n d e d b y t h e

c o m p e t e n t i n t e r n a t i o n a l s c i e n t i f i c b o d i e s

T h e a f f i l i a t i o n s o f a u t h o r s a r e t h o s e g i v e n a t t h e t i m e o f n o m i n a t i o n .

T h e u s e i n t h e s e P r o c e e d i n g s o f p a r t i c u l a r d e s i g n a t i o n s o f c o u n t r i e s o r

t e r r i t o r i e s d o e s n o t i m p l y a n y j u d g e m e n t b y t h e A g e n c y a s to t h e l e g a l s t a t u s

o f s u c h c o u n t r i e s o r t e r r i t o r i e s , o f t h e i r a u t h o r i t i e s a n d i n s t i t u t i o n s o r o f

t h e d e l i m i t a t i o n o f t h e i r b o u n d a r i e s .

T h e m e n t i o n o f s p e c i f i c c o m p a n i e s o r o f t h e i r p r o d u c t s o r b r a n d - n a m e s

d o e s n o t i m p l y a n y e n d o r s e m e n t o r r e c o m m e n d a t i o n o n t h e p a r t o f t h e

I n t e r n a t i o n a l A t o m i c E n e r g y A g e n c y .

CONTENTS OF VOL. II

G eom etric v aria tio n a l p rob lem s from a m easu re -th e o re ticpoint of view (IA E A -SM R -11/9) ..................................................................... 1F . J . A l m g r e n , J r .

A rea m e a su re s on a r e a l vecto r space (IA E A -SM R -11 / 10) ................ 23F . B r i c k e l l

The d ifferen tiab ility of tran sfo rm atio n s which p re se rv e g eo d esic s(IA E A -SM R -11 / 11) ................................................................................................. 33F . B r i c k e l l

Stability th eorem s fo r R 2-action s on m anifo lds (IA E A -SM R -11/12) .. 37C. C a m a c h o

Introduction to m in im a l-su rface theory (IA E A -SM R -11 / 13) ................ 43E . D e G i o r g i

On com plex v a r ie t ie s of nilpotent L ie a lg e b ra s (after G. F av re )(IA E A -SM R -11/14) ................................................................................................. 47P . D e l a H a r p e

On in fin ite-d im en sion al L ie groups acting on fin ite-d im en sion alm anifo lds (IA E A -S M R -ll/1 5 ) ........................................................................... 59P . D e l a H a r p e

Some p ro p e rtie s of in fin ite-d im en sion al orthogonal groups(IA E A -SM R -11/16) ................................................................................................. 71P . D e l a H a r p e

T heory of r e s id u e s in se v e ra l v a r ia b le s ( I A E A - S M R - ll /17) ............... 79P . D o l b e a u l t

On connections (IA E A -SM R -11 / 18) ................................................................... 97P . D o l b e a u l t

Introduction to global calcu lu s of v a r ia t io n s (IA E A -SM R -11 / 19) ....... 113H.I. E 1 i a s s о n

E lem en tary survey of p seu do -d ifferen tia l o p e ra to rs and thew ave-front se t of a d istribution (IA E A -SM R -11 /20) ........................... 137R . J . E l l i o t t

Boundary value p rob lem s fo r non -lin ear p a r t ia l d ifferen tia lequations (IA E A -SM R -11/21 ) ........................................................................... 145R .J . E l l i o t t

G au ssian m e a su re s on Banach sp a c e s and m anifolds(IA E A -SM R -11/22) ................................................................................................ 151K.D . E 1 w o r t h у

Sheaf cohom ology, s tru c tu re s on m anifo lds and vanish ing theory(IA E A -SM R -11/23) ................................................................................................ 167M .J. F i e l d .

Com plex an a ly s is ori Banach sp a c e s (IA E A -S M R -ll/2 4 ) ........................ 18ЭM .J. F i e l d

M odern theory of b ill ia rd s - an introduction (IA E A -SM R -11 /2 5 ) ........ 193G. G a l l a v o t t i

Som e re m a rk s on q u asi-A b elian m anifolds (IA E A -SM R -11/26) .......... 203F . G h e r a r d e l l i , A. A n d r e o t t i

L ife and death of the B ern ste in problem (IA E A -SM R -11 / 27) ................ 207E . G i u s t i

In v arian ts of fo lia tio n s (IA E A -SM R -11/28) ................................................... 215C. G o d b i l l o n

On the lo ca l so lvab ility of lin ear p a r tia l d iffe ren tia l equations(IA E A -SM R -11 /29) .................................................................................................. 221H. G o l d s c h m i d t

Com pact o p e ra to rs and the m inim ax princip le (IA E A -SM R -11 / 30) . .. 225R .A . G o l d s t e i n , R. S a e k s

R igid ity and en ergy (IA E A -SM R -11 / 31) ........................................................ 233R .A . G o l d s t e i n , P . J . R y a n

P h ase tran sitio n s in D -dim ension al Ising la tt ice s(IA E A -SM R -11/32) .................................................................................................. 245R .A . G o l d s t e i n , J . J . K o z a k

D ifferen tia l calcu lu s in lo ca lly convex sp ac e s (IA E A -SM R -11 / 33) . .. 263R .A . G r a f f

S in gu laritie s in "so a p bubbles" and "so a p film s"(IA E A -SM R -11 / 34) ................................................................................................. 271Je a n E . T a y l o r

C au stic s (IA EA -SM R -11/35) .................................................................................. 281J . G u c k e n h e i m e r

The top o log ical degree on Banach m anifolds (IA E A -SM R -ll/3 6 ) ....... 291C .A .S . I s n a r d

E x isten ce and n on-existence fo r se m i- lin e a r e llip tic equations(IA E A -SM R -11 /37) ................................................................................................. 315J .L . K a z d a n

An exam ple of a stran ge th ree-d im en sio n al su rfac e in C2(IA E A -SM R -11 /38) ................................................................................................. 323J . J . K o h n

A continuous change of topo logical type of R iem annian m anifolds and it s connection with the evolution of harm onic fo rm s andspin s tru c tu re s (IA E A -SM R -11/39) ............................................................. 329J . K o m o r o w s k i

A new proof fo r re g u la r ity of so lu tions of e llip tic d iffe ren tia lo p e ra to rs (IA E A -SM R -11/40) ........................................................................... 355M. K u r a n i s h i

S e c re ta r ia t of Sem in ar ............................................................................................ 363

lA EA -SM R -11/9

GEOMETRIC VARIATIONAL PROBLEMS FROM A MEASURE-THEORETIC POINT OF VIEW

F.J. ALMGREN, Jr. *Department of Mathematics,Princeton University,Princeton, N.J.,United States of America

Abstract

GEOMETRIC VARIATIONAL PROBLEMS FROM A MEASURE-THEORETIC POINT OF VIEW.Some phenomena of geometric variational problems are treated; in particular, a discussion of surfaces

as measures, a regularity theorem, and estimates on singular sets are presented.

"D uring the la s t three d ecad es the su b jec t o f geo m etric m e asu re theory has developed from a co llection of iso la ted sp e c ia l r e su lts into a coh esive body o f b a s ic knowledge with an am ple n atu ral s tru c tu re of it s own, and with stro n g t ie s to m any other p a r ts o f m a th e m a tic s .T h ese advan ces have given us d eep er p erception o f the an alytic and top o log ica l foundations of geom etry , and have provided a new d irection to the ca lcu lu s o f v a r ia t io n s . R ecently the m ethods o f geo m etric m e a su re theory have led to v ery su b sta n tia l p r o g r e s s in the study of quite g e n era l e llip tic v a r ia t io n a l p ro b lem s, including the m u ltid im en sio n al p rob lem of le a s t a re a . " [ 1 ( P r e fa c e ) ] .

T h is a r t ic le , con sistin g o f five p a r t s , is intended a s an introduction to the co llection o f m ath em atica l techniques and r e su lt s known a s geo m etric m e a su re theory from the point of view of ce rta in p ro b lem s a r is in g in the calcu lu s o f v a r ia t io n s .

P A R T A. SOM E PHENOM ENA O F GEOM ETRIC VARIATIONAL PR O BLEM S

Suppose k s n a re p o sitiv e in tegers and one is given a reaso n ab le su r fa c e S o f d im ension к in R n. A ssu m e a lso one i s g iven a su itab le function

F : Rn X G(n, k) - R+

H ere G(n, k) denotes the G ra ssm a n n m anifold o f a ll unoriented k -d im en sion al p lan es through the o r ig in in Rn o r , equivalently , the sp ace o f a ll k-d im ensional

The preparation of this article was supported in part by a grant from the National Science Foundation and in part by funds from the Science Research Council in connection with the Symposium on Global Analysis, 1971 - 1972, at the University of Warwick.

1

2 ALMGREN

tangent plane d irection s in Rn. Then one can define the in teg ra l F (S) o f F ov er S by the form ula

Z(S) = J F (p ,T an k(S ,p )) dg?k pp £ S

H ere Tank(S, p )e G(n, k) denotes the k -d im en sion al tangent plane d irection to S at p fo r р б S , and.gfk denotes the k-d im en sion al H au sdorff m e asu re o v er R n defined in C . 3(1). The k-d im en sion al H au sdorff m e a su re o f a k -d im en sion al subm anifold M of Rn of c la s s 1 a g re e s with any other re a so n able definition o f the к- a r e a o f M. H ow ever, with the use o f the H ausdorff к-m e a su re , one can m ake m ath em atically p r e c is e the notion o f the k- d im en sion al a r e a on su r fa c e s which m ay have e s se n t ia l s in g u la r it ie s .1

With th is term inology, there a re s e v e r a l p ro b lem s one m ight w ish to study.

F i r s t prob lem (ex isten ce): Among a ll su r fa c e s S having, say , a p r e sc r ib e d boundary В (and p o ssib ly sa tis fy in g other co n stra in ts), is there one m inim izing F (S)?

Second prob lem (regu larity ): If there is a solution to the f i r s t problem , how nice i s it? In p a r t ic u la r , is it gen era lly like a sm ooth k -d im en sion al subm anifold o f R11?

T h ird prob lem (stru c tu re o f s in g u la r itie s ) : What kind o f s in g u la r itie s a re p o ss ib le in so lu tion s to the f i r s t p rob lem (if any)? And if s in g u la r itie s a re p o ss ib le , what language should one use to d e sc r ib e them ?

Fourth p rob lem (com putation): If there a r e so lu tion s to the f i r s t p rob lem , how does one exp lic itly find them ?

To m ake th ese p ro b lem s m ath em atically p re c ise , th ere a re o f co u rse s e v e r a l qu estio n s to be se ttled , nam ely:

(1) What is a su r fa c e ?(2) What i s the boundary o f a su r fa c e ?(3) What a re rea so n ab le conditions to put on F ? (We a lread y have been

assu m in g im p lic itly that F is m e asu rab le , fo r exam ple . )

B e fo re attem pting to answ er th ese qu estio n s, it i s p erh ap s u sefu l to co n sid er the phenom ena which a r i s e when one stu d ie s th ese p ro b lem s.In the follow ing ex am p le s, the in tegrand F is r e s tr ic te d to being iden tically 1, i . e . fo r each (p, ж) e Rn X G(n, k), F (p , ir) = 1. The problem of m inim izing F (S) thus becom es the p rob lem o f m inim izing £fk(S), in other w ords, the

1 There are many distinct k-dimensional measures over Rn. Although all of them are Borel regular, invariant under the Euclidean group, defined by reasonable geometric constructions, and agree with the Hausdorff measure on class 1 submanifolds, they can disagree completely on more general subsets of Rn.One reason for using the Hausdorff measure lies in the structure theorem for sets of finite Hausdorff measure [1 (3 .3 .1 3 )] which says that any Borel subset A of Rn for which ^ ( A ) <*> can be written almost uniquely as A = В U С , В П С = 0, such that В can be covered by a countable fam ily of k-dimensional submanifolds of Rn of class 1 and S ^ ttC ) = 0 for almost all orthogonal projections эт: Rn -» R^; h e reS ^ denotes Lebesgue k-dimensional measure over R^, which e q u a l s o v e r R , This structure theorem, proved for к = 1, n = 2 by A .S . Besicovitch and in general dimensions by H. Federer, is one of the several main achievements of the geometric measure theory, and, typically, it plays an essential role in proving the existence of solutions to variational problems in the measure-theory context.

IA EA -SM R-11/9 3

FIG.l. A disk with five handles.

p ro b le m of m in im iz in g th e к -d im e n s io n a l a r e a o f S. The v a r io u s p ro b le m s a s s o c ia te d w ith m in im iz in g a r e a a re so m e tim e s ca lle d c o lle c tiv e ly P la te a u 's P ro b le m in honour o f the B e lg ia n p h y s ic is t J . P la te a u of the la s t ce n tu ry who, am ong o th e r th in g s , s tu d ied the g e o m e try of soap f ilm s and soap bubb les (see [2 ] ) .

E x am p le 1. Suppose С is a fixed s im p le c lo sed c u rv e in R3 of f in ite leng th and le t Do denote a fixed tw o -d im e n s io n a l d isk , say D0 = R2n {(x ,y): x 2 + y 2 S 1}. T he f i r s t p e r s o n to m ake s ig n ific an t p r o g re s s in a s p e c ia l fo rm u la tio n of the p ro b le m of le a s t tw o -d im e n s io n a l a r e a w as the la te J . D oug las who show ed in p a r t ic u la r :

T h e o re m [3]: Am ong a ll m ap s D0 ->■ R3 su ch th a t 3D0 m aps hom eo- m o rp h ic a lly onto C , th e re e x is ts a m app ing of le a s t tw o -d im e n s io n a l a r e a (" tw o -d im e n s io n a l a r e a " as u sed in th is th e o re m is defined e x p lic itly in B . 6).

U n fo rtu n a te ly , th e re a r e no known d i r e c t ex ten s io n s of th is r e s u l t to h ig h e r d im e n s io n a l d isk s .

Now fo r m = 1 ,2 ,3 , . . . le t Dm denote a fixed tw o -d im e n s io n a l d isk w ith m hand les (a D5 a p p e a rs in F ig . 1), and le t Am deno te th e in fim um of the a r e a s of th e m app ings Dm -» R 3 su c h th a t 3Dm m ap s h o m eo m o rp h ica lly onto С (m appings re a liz in g th is in fim u m a r e a do no t s e e m to be known to e x is t) . S ince one can alw ays "p inch out" a hand le b e fo re m apping Dm , one h as th e obvious in e q u a litie s

A 0 ê A , = A2 g . . . й l im m Am g 0.

F o r a cu rv e lik e th a t sk e tch e d in F ig . 2 (a s im p le c lo sed unknotted cu rv e of f in ite leng th ), W .H . 'F le m in g has p ro v ed the s t r i c t in e q u a litie s

A 0 > A , > A2 > . . . > l im m Am > 0 [4 ] .

T he s ig n ific a n c e o f th e s e in e q u a li tie s is th a t if one w ish e s to so lv e the p ro b le m of m in im iz in g a r e a am ong a l l o r ie n te d su r fa c e s having boundary C, th e n one canno t find an ab so lu te m in im u m am ong o r ie n te d s u r fa c e s o f f in ite to p o lo g ica l ty p e , s in c e , indeed , the l i s t o f Dm 's is a co m p le te to p o lo g ica l c la s s if ic a t io n of a l l co m p ac t o r ie n ta b le tw o -d im e n s io n a l m an ifo ld s having a c i r c le a s boundary . O n th e o th e r hand, th e re is a su r fa c e S (sk e tch ed in F ig . 3) w hich p e rh a p s d e s e rv e s to be ca lle d th e o r ie n te d s u r fa c e of le a s t

4 ALMGREN

FIG.3. The oriented surface S having boundary С and being of least area is of infinite topological type.

a r e a having С a s bou n d ary . S ~ С is a tw o -d im e n s io n a l r e a l an a ly tic sub m an ifo ld of R3 having 0 m e an c u rv a tu re a t each poin t and the a r e a of S eq u a ls l im m A m (see E . 1).

The s u r fa c e S is o f in fin ite to p o lo g ica l type and is hom eo m o rp h ic w ith the s u r fa c e T sk e tch e d in F ig . 4. (The n a tu re of th is s u r fa c e T su g g e s ts why, f re q u en tly , h o m o lo g ica l cond itions in g e o m e tr ic m e a s u re th e o ry a r e s ta te d in te rm s of the V ie to r is o r C ech th e o r ie s r a th e r th an the s in g u la r th e o ry ).

E x am p le 2. S uppose В c o n s is ts of th re e p o in ts in R2 w hich a re the v e r t ic e s o f an e q u i la te ra l t r ia n g le w ith c e n tre a t 0 e R2 . T hen am ong a ll o n e -d im e n s io n a l s e ts S w hich a r e the unions of n o n - tr iv ia l re c tif ia b le a r c s and th ro u g h w hich each po in t o f В is pa thw ise connected to ea ch o th e r po in t, the unique s e t Y (see F ig . 5) o f le a s t to ta l leng th c o n s is ts of the union of the th r e e lin e se g m e n ts connecting the p o in ts of В to 0 (the p ro o f of th is is not co m p le te ly t r iv ia l) . T h is p ro b le m is a s n a tu ra lly posed as one could ask , the in te g ra l (length) is r e a l an a ly tic , the boundary В is a lg e b ra ic , and the so lu tio n Y is unique. N e v e r th e le s s , the so lu tio n has an in te r io r s in g u la r ity of co d im en s io n 1, n am ely 0 e Y с R 2 w h ere the th r e e lin e se g m en ts m e e t.

E x am p le 3. S uppose V is a com plex a lg e b ra ic v a r ie ty of com plex d im e n sio n к in com plex n -d im e n s io n a l sp a ce (En, and le t U c Œn be open and bounded su ch th a t 9(V n U ) is su ita b le (in the language of C .3 (4 ), V nU is re q u ir e d to be a 2 k -d im e n s io n a l in te g ra l c u r r e n t in R = Œn); the com plex s t r u c tu re of V, of c o u rs e , g iv es V a n a tu ra l o r ie n ta tio n so th a t V n U to g e th e r w ith th a t o r ie n ta tio n b eco m es an o r ie n te d 2k) r e c tif ia b le and^ ^ k-m e a s u ra b le s u b se t o f R 2n [ 1(4. 2 .2 9 )] . L e t W be any o th e r o r ie n te d

IA EA -SM R-11/9 5

FIG.4. A disk T with infinitely many handles converging to a boundary point.

FIG. 5. The 1-dimensional set Y connecting the three points of the boundary В and of least length has an interior singularity of codimension 1.

2 k -d im e n s io n a l su r fa c e ( in te g ra l c u r re n t) such th a t 3W = 9(V n U). Then e i th e r W = V n U o r the 2 k -a re a of W (M(W)) is s t r ic t ly l a r g e r th an the 2 k -a re a o f V (lU (M(V n U)). F u r th e rm o re , th is is t r u e w h e th e r o r not V has s in g u la r i t ie s . In p a r t ic u la r , if one w ish e s to so lv e th e p ro b le m of m in im iz in g o r ie n te d a re a , and r e a l ly ach iev e the le a s t a r e a , th en one m u s t a t t im e s ad m it á s s in g u la r i t ie s in the so lu tio n s to P la te a u 's p ro b le m a t le a s t a ll the s in g u la r i t ie s o cc u rin g in co m p lex a lg e b ra ic v a r ie t ie s .

E x am p le 4. R . T hom has c o n s tru c te d a 1 4 -d im e n s io n a l co m p ac t r e a l a n a ly tic m an ifo ld M w ithou t boundary w ith a s e v e n -d im e n s io n a l in te g ra l hom ology c la s s и w hich canno t be r e p re s e n te d by any se v e n -d im e n s io n a l sm o o th subm an ifo ld [ 5] . On the o th e r hand e v e ry in te g ra l hom ology c la s s of any co m p ac t sm o o th R iem an n ian m an ifo ld can be r e p re s e n te d by an o r ie n te d s u r fa c e ( in te g ra l c u r re n t) o f le a s t a r e a (m ass) in th a t c la s s .T h o m 's exam ple th u s show s th a t so m e tim e s th e re can be topo log ica l o b s tru c tio n s to s u r fa c e s o f le a s t a r e a being f re e of s in g u la r i t ie s .

E x am p le 5. C o n s id e r the follow ing p a r titio n in g p ro b le m . L e t т х, т г , . . . , т 4> 0. T hen am ong a l l d is jo in te d re g io n s A 1; . . . , A , c R “ su ch th a t ¿^ (A j) = m¡ fo r each i, a r e th e re re g io n s fo r w hich

9 A i)i = 1

a t ta in s a m in im u m v alu e? T h is p ro b le m alw ays ad m its so lu tio n s [6 ] and g e n e ra l a rg u m e n ts (see D .3 ) show th a t ex cep t fo r a (p o ssib ly em pty) com pact

6 ALMGREN

FIG. 6. The unique minimal partitioning configuration for two regions of prescribed volumes.

FIG. 7. A "soap-bubble-like” minimal partitioning configuration consisting of six real analytic surfaces meeting at 120° along four smooth arcs which meet at equal angles at two points.

s in g u la r s e t of z e ro ^ n-1 m e a s u re , U ^ A j is a H o lder con tinuously d if fe r - en tiab le n - 1 d im e n s io n a l subm an ifo ld of R n. The p a r t ic u la r fo rm of the p ro b le m above f u r th e r im p lie s the r e a l a n a ly tic ity o f the r e g u la r p a r t of Ui8Ai. F o r i = 1, the unique so lu tio n to th is p ro b le m (up to is o m e tr ie s of R n, of c o u rse ) is a s ta n d a rd n -b a ll o f th e p r e s c r ib e d v o lu m e . F o r i = 2, n = 3, the unique so lu tio n (sk e tch ed in F ig . 6, a lso sk e tch e d "blow n a p a r t" ) c o n s is ts of th r e e s p h e r ic a l p ie c e s m ee tin g along a c i rc le (in th is c a se the c i r c le is the co m p ac t s in g u la r s e t o f z e ro Çf2 m e a s u re r e f e r r e d to above and in D. 3). F o r t = 3, n = 3, the so lu tio n se e m s to be th a t sk e tch e d in F ig . 7; no te th a t s ix p ie c e s of r e a l a n a ly tic s u r fa c e m e e t ta n g en tia lly a t 120° along fo u r sm o o th a r c s w hich in tu rn m e e t a t two v e r t ic e s ta n g en tia lly as the c e n tr a l cone o v e r the v e r t i c e s o f a r e g u la r te tra h e d ro n (see [ 2] ).

IA EA -SM R-11/9 7

FIG. 8. The Môbius band as a surface of least area.

FIG.9. The triple Mflbius band as a surface of least area.

E x am p le 6. A M obius b a n d -lik e s u r fa c e (sk e tch ed in F ig . 8) o c c u rs as a soap film fo r a w ire ben t in the shape of the boundary shown, and a lso o c c u rs as a s u r fa c e o f l e a s t a r e a am ong a l l m a th e m a tic a l s u r fa c e s spanning sue!} a boundary in th e s e n se o f hom ology w ith co e ffic ie n ts in the in te g e rs m odulo 2 (see E .2 ) . S im ila r ly a t r ip le M obius b an d -lik e s u r fa c e (ske tched in F ig . 9) o c c u rs a s a soap film fo r a w ire ben t in the sh ap e of th e boundary show n, and a lso o c c u rs as a s u r fa c e of le a s t a r e a am ong a l l m a th e m a tic a l s u r fa c e s w hich sp an su ch a boundary in the s e n se o f hom ology w ith c o e ffic ie n ts in the in te g e rs m odulo 3 (see E .3 ) .

F in a lly a s u r fa c e S lik e th a t sk e tch e d in F ig . 10 (like a M obius band on th e le f t jo ined to a t r ip le M obius band on th e r ig h t by a th in rib b o n of s u r fa c e , having as b oundary С a s in g le s im p le c lo sed unknotted cu rv e ) o c c u rs both a s a soap film and as a m a th e m a tic a l m in im a l s u r fa c e . H ow ever, J . F .A dam s h as po in ted out the e x is te n c e of a con tinuous r e t r a c t io n (S, C) -* (С, C) of S onto the b oundary С [ 7 (Appendix)] so th a t, in no w ay, in the s e n se of a lg e b ra ic topology , does S "sp an " C . In p a r t ic u la r , if one w ish e s to r e g a rd

8 ALMGREN

FIG. 11. A soap film with a boundary wire which is not closed.

C as the boundary of S, th en one m u s t use o th e r d e fin itio n s of boundary of a s u r fa c e th a n th o se of a lg e b ra ic topology (see , in p a r t ic u la r , th e v a r ia t io n a l fo rm u la tio n in [ 8] ).

E x am p le 7. S uppose one bends a w ire in to the shape of an o v erhand knot as sk e tch ed in F ig . 11(a) (note th a t the two ends o f the w ire a r e f re e ) . T y p ica lly , w hen su c h a w ire i s d ipped in soapy w a te r , a film su ch as th a t

IA EA -SM R-11/9 9

sk e tch e d in F ig . 11(b) fo rm s , even though the w ire is not c lo se d . Such a film does ad m it a m a th e m a tic a l ap p ro x im atio n , but only w ith a "boundary" o f su b s ta n tia l p o s itiv e th ic k n e s s . Indeed one can p ro v e by ta n g en t cone a rg u m e n ts (see the n ice d is c u s s io n of su ch cones in R ef. [ 2] ) th a t such a m a th e m a tic a l s u r fa c e is im p o ss ib le o v e r an in fin ite ly th in b oundary of c la s s 3. The s ig n ific an c e of th is , am ong o th e r th in g s , is th a t if one w ish e s to c o n s tru c t a th e o ry of m in im a l s u r fa c e s w hich in p a r t ic u la r in c lu d es the phenom ena su g g e s ted by soap f ilm s , th en one m u s t a t t im e s ad m it b o u n d a rie s of su b s ta n tia l p o s itiv e th ic k n e s s .

E x am p le 8. Suppose В = {(-2, 0), (2, 0)} c R 2. T hen am ong a l l oned im e n s io n a l s e ts S ly ing in R 2 n {(x, y): x2 + y2 S 1} , w hich a r e unions ofn o n - tr iv ia l r e c tif ia b le a r c s th ro u g h w hich the p o in ts of В a r e pa thw ise co n n ected to ea ch o th e r , th e re a r e ex a c tly tw o d is t in c t s e ts С and С ' of le a s t to ta l leng th , w h ere C- = L 1U L 2U L 3,

L x = {(x, у): у = 3'* (2 + x) , -2 S x á -3^}

4 = {(x,y): у = 3_i (2 - x) , 31 i x é 2}

L 3 = {(x, у): у = (1 - x 2 )", - 3 ‘* â x â 3}

and С 1 is th e im ag e of С u nder re f le c tio n a c ro s s the x -a x is (see F ig . 12). T h is p ro b le m is n a tu ra lly po sed , the in te g ra l (length) is r e a l an a ly tic , the boundary В and the o b s ta c le {(x, y): x 2 + y2 < 1} a r e a lg e b ra ic , and th e re a r e ex a c tly two so lu tio n s (a t r iv ia l m o d ific a tio n m a k es th e so lu tio n unique). N e v e r th e le s s each so lu tio n c u rv e , a lthough a o n e -d im e n s io n a l subm anifo ld of R2 of c la s s 1, is no t a sub m an ifo ld o f c la s s 2. T he ta n g en t lin e s of С and C, how ever, do v a ry in a L ip sc h itz ia n m a n n e r , hence a f o r t io r i H o lder con tinuously (see D .3 ) .

E x am p le 9. S uppose В = { (0 ,-1 ) , (0, 1)}. T hen the unique o n ed im e n s io n a l s e t С in R 2 of le a s t leng th connecting the p o in ts o f В is , of c o u rse , С = R2 n {(x, у ) : x = 0, -1 s y û 1} (see F ig . 13(a)). Now le t f: R2 -► R 2 be the a lg e b ra ic d iffe o m o rp h ism g iv en by f(x, y) = (x + y 3, y).

FIG. 12 . The curve С o f le a s t len g th is not o f class 2.

10 ALMGREN

The len g th in te g ra n d F : R2 X G(n, 1) -* {1} tr a n s f o r m s n a tu ra lly under f to g ive a new in te g ra n d G = f#F w ith the obvious p ro p e r ty th a t the unique o n ed im e n s io n a l s e t D of le a s t G in te g ra l am ong th o se s e ts connecting the po in ts o f f(B) is D = f(C ). N ote th a t D is a lso the g ra p h of the function y = fo r -1 û x § 1 (see F ig . 13(b)). The e llip tic ity (D. 1(1)(2)) o f F im p lie s the e l lip tic ity o f G, and i t is c le a r th a t the function y = x^ is the unique n a tu ra l so lu tio n to th e r e a l an a ly tic e l lip tic E u le r -L a g ra n g e d if fe re n tia l equation a s s o c ia te d w ith G and th e s ta n d a rd (x, y) c o o rd in a te s fo r R 2 (see D .2 ) .T h is function is no t even of c la s s 1, how ever (although it is r e a l an a ly tic ex cep t fo r a co m p ac t s in g u la r s e t of z e ro m e a s u re — a r e p re s e n ta t iv e co n c lu sio n fo r su ch p ro b le m s (see C . l(5)(d))).

P A R T B . GEOM ETRIC VARIATIONAL PRO BLEM S IN A M APPING SETTING AND ASSOCIATED VARIFOLDS

В . 1. V a r ia tio n a l p ro b le m s in a m apping se ttin g

S uppose one is g iven a su ita b le open s e t W in Rk. We w ill denote by „<?an a p p ro p r ia te sp a ce of m app ings f: c lo s u re W -* Rn, p e rh a p s w ith f |9W p r e s c r ib e d . W e w ill su p p o se a lso we a r e g iven a re a so n a b le function

IA EA -SM R-11/9 U

Ф: -► R . In c a se к = 0, n = 1, th e n Л can be re g a rd e d a s , sa y , the sp a ceof c la s s 1 m app ings Ф: R -» R, and the b a s ic p ro b le m w hich led to the d if fe re n tia l ca lc u lu s w as th a t o f finding a po in t w h ere Ф a s s u m e s i ts m a x im um v a lu e . E q u iv a len tly one could se e k th o se po in ts w h ere Ф ta k e s i ts m in im u m v a lu e , o r , m o re g e n e ra lly , one could se e k c r i t i c a l p o in ts o f Ф, i. e . p o in ts a t w hich the f i r s t d e r iv a tiv e Ф' v a n ish e s . In c a se k â 1, n i 1, then , h e u r is t ic a lly a t le a s t , the b a s ic p ro b le m of th e ca lc u lu s of v a r ia tio n s (in th is con tex t) is th a t o f finding p o in ts (ac tu a lly m app ings f: c lo s u re W -> Rn) a t w hich (m o st com m only) Ф a s s u m e s i t s m in im u m v a lu e , o r , m o re g e n e ra lly , the c r i t i c a l p o in ts of Ф, i . e . w h ere the f i r s t v a r ia t io n 6Ф v a n ish e s id e n tic a lly (in p ra c t ic e the d e fin itio n o f f i r s t v a r ia t io n v a r ie s c o n s id e ra b ly f ro m p ro b le m to p ro b le m ). In a n u m b e r of w ays, as the above p h ra se o lo g y su g g e s ts , th e re a r e an a lo g ie s be tw een c a lc u lu s and the ca lc u lu s of v a r ia t io n s . The o rd in a ry ca lc u lu s has b een ex tended to d if fe re n tia l m an ifo ld s in v a r io u s fa sh io n s , and it is so m e tim e s u se fu l to r e g a rd c e r ta in p ro b le m s in the ca lc u lu s of v a r ia t io n s in th e language o f the o rd in a ry ca lc u lu s , but ex tended to m an ifo ld s having in fin ite d im e n s io n s . T h ese m a n ifo ld s o f in fin ite d im e n sio n ty p ic a lly a r e "m o d e lled on" H ilb e r t sp a c e s o r B anach s p a c e s . U n fo rtu n a te ly , su c h in fin ite d im e n s io n a l m an ifo ld s so f a r have not p layed a s ig n if ic a n t ro le in the g e o m e tr ic v a r ia t io n a l p ro b le m s w ith w hich we a re m a in ly c o n c e rn ed h e re .

B . 2. S p ec ia l fo rm s o f th e function Ф

The functions Ф;„<^ -► R w hich have re c e iv e d the g r e a te s t m a th e m a tic a l a tten tio n in h ig h e r d im e n sio n s a r e in te g ra ls o f th e fo rm

(*) Ф(Г) = J <p(x, f(x), Df(x)) d x x d x 2 . . . dxk, f e . . ^ x s w

H e re <p: W X RnX Hom(Rk,R n) -► R+ is the in te g ra n d a s so c ia te d w ith Ф.F o r ex am p le , th e D ir ic h le t in te g ra n d <pD is g iven by

к к

<pD(x, f(x), Df(x)) = IIDf(x) ¡I2 [ O f J/a x i ) ( x ) ]2

i = l j = l

A nother ex am p le is the k -d im e n s io n a l a r e a in te g ra n d cpA w hich is defined by se ttin g

cpA (x, f(x), Df(x)) = II Лк Df(x) I

H e re AkDf(x): A kRk -> AkRn and ||A kD f(x)|| is eq u a l to the s q u a re ro o t of the su m of th e s q u a re s o f a l l th e к by к m in o rs of the n by n Ja co b ia n m a tr ix

12 ALMGREN

9 f 19x- (x)

9 f2 , . 9 Z (X)

9 fnЭх-, (x)

9 f 19 x 2

9 f 2 9x0

9 fn3xn

(x)

(X)

(X)

df}Эх,

9fЭх.,

(x)

(X)

9f9xi (x)

As in d ic a ted , we a r e p r im a r i ly co n c e rn ed w ith g e o m e tr ic in te g ra ls Ф c o rre sp o n d in g to g e o m e tr ic in te g ra n d s q>. S e v e ra l d e fin itio n s a re in o rd e r .

B .3 . D efin itions

(1) A function Ф: Л -+ R+ is ca lled an in te g ra l in p a r a m e tr ic fo rm if and only if Ф has the fo rm (*) above and, in add ition , fo r each fç.jX and ea ch d if fe o m o rp h ism g: W -» W,

0(f) = S (fo g )

(2) A function <p: W X Rn X Horn (Rk, Rn) -» R is ca lle d an in te g ran din p a r a m e tr ic fo rm if and only if the function Ф: R defined by (*) isan in te g ra l in p a r a m e tr ic fo rm .

(3) A function Ф: R+ is ca lled a g e o m e tr ic in te g ra l if and onlyif Ф has th e fo rm (*) and, in add ition , th e re e x is ts a function

F: Rn X G(n, k) - R+

su c h th a t fo r each f e .

$(f) = J F [ p , T a n k(f(W ), p)] N (f,p ) d ^ r k pp e f ( W )

H ere N(f, p) = c a rd f 1{p} fo r ea ch p e f (W ) .(4) A function i p : W X R n X H om (Rk , Rn ) -» R is ca lle d a g e o m e tr ic

in te g ra n d if and only if the function Ф defined by (*) is a g e o m e tr ic in te g ra l .

В . 4 . R e m a rk

To the b e s t of m y know ledge the e x p re s s io n " in te g ra l in p a ra m e tr ic fo rm " w as in tro d u c e d by C .B . M o rre y , J r . to su g g e s t th a t th e a p p ro p ria te in te g ra ls to be s tu d ied o v e r p a r a m e tr ic su r fa c e s (su rfa c e s w hich a r e not

IA EA -SM R-11/9 13

th e g ra p h s of functions) a r e in te g ra ls in p a r a m e tr ic fo rm [ 9 ] . T h is t e r m i nology has b een a s o u rc e of confusion , how ever, s in c e , in p a r t ic u la r , in te g ra ls in p a r a m e tr ic fo rm a re p r e c is e ly th e in te g ra ls of fo rm (*) w hich do not depend on the p a ra m e tr iz a t io n of th e dom ain .

F o r su ffic ie n tly n ice f it is c le a r th a t any g e o m e tr ic in te g ra l has the in v a ria n c e p ro p e r ty c h a r a c te r is t ic of in te g ra ls in p a r a m e tr ic fo rm .In c a se the m a p s in Л a r e L ip sc h itz ia n and T ank (f(W ),p) is u n d ersto o d to m e an the (,g^kL f(W ), k) a p p ro x im a te tan g en t cone, T an k( ^ kL_ f(W), k) [ 1 ( 3 .2 .1 6 ) ] , th en any g e o m e tr ic in te g ra l is indeed an in te g ra l in p a r a m e tr ic fo rm . A lso , in c a se Ф is an in te g ra l in p a r a m e tr ic fo rm w hich is of c la s s 2, ra n g e $ c R +, and Ф is independen t o f o r ie n ta tio n , th e n C .B . M o rre y , J r . has show n th a t $ is a g e o m e tr ic in te g ra l [ 9 ].

In te g ra ls in p a r a m e tr ic fo rm , and g e o m e tr ic in te g ra ls in p a r t ic u la r , a r e of sp e c ia l g e o m e tr ic s ig n ific an c e s in c e the value of the in te g ra l depends only on the g e o m e try of f(W) and not on the p a r t ic u la r p a ra m e tr iz a tio n w hich p ro d u c e s i t . I know of no n a tu ra l g e o m e tr ic , p h y s ic a l, o r b io lo g ica l p ro b le m s involving in te g ra ls in p a r a m e tr ic fo rm w hich a r e not a lso g e o m e tr ic in te g ra ls .

The к -d im e n s io n a l a r e a in te g ran d is an exam ple of a g e o m e tr ic in te g ran d w hile the D ir ic h le t in te g ran d is n e i th e r a g e o m e tr ic in te g ran d n o r an in te g ran d in p a r a m e tr ic fo rm .

One r e f e r s to a v a r ia t io n a l p ro b le m a s being a g e o m e tr ic v a r ia t io n a l p ro b le m in c a s e the only in te g ra ls w hich a r i s e in the p ro b le m a r e g e o m e tr ic in te g ra ls .

B . 5. Som e d iffic u ltie s w ith g e o m e tr ic v a r ia t io n a l p ro b le m s

T he g r e a te s t in it ia l d ifficu lty w ith the s tudy of g e o m e tr ic v a r ia t io n a l p ro b le m s (in the m apping se ttin g ) is th a t, in g e n e ra l, th e re is no s t r a ig h tfo rw a rd way to ob ta in so lu tio n s by th e so ca lle d d ir e c t m ethod of the ca lc u lu s of v a r ia t io n s . The te r m " d i r e c t m ethod o f th e ca lc u lu s of v a r ia t io n s " is used lo o se ly in a n u m b e r of d if fe re n t co n tex ts . Roughly speak ing the " d i r e c t m ethod" m a k es se n se in a v a r ia t io n a l p ro b le m w hen the fo rm u la tio n of the p ro b le m i ts e l f g u a ra n te e s th e ex is te n c e of a seq u en ce {fjlj of m app ings in „ ^ , any su ita b le co n v e rg en t (in„<f) su b seq u en ce of w hich would y ie ld a so lu tio n to the p ro b lem ; fo r ex am p le , if one se e k s a m in im u m fo r Ф, one m igh t have l im ¡$ ( f i ) = in f Ф. The d ir e c t m ethod of the c a lc u lu s of v a r ia tio n s is ap p lica b le w hen se q u en c es {f^j a s above do in fa c t ad m it su b se q u en c es co n v e rg en t to so lu tio n s .

The m a in d if f ic u ltie s w hich I know about in th e a tte m p t to use th e d ir e c t m ethod in g e o m e tr ic v a r ia t io n a l p ro b le m s se e m b e s t i l lu s tr a te d by ex a m p le s .

B .6 . E x am p les

Suppose к = 2, n = 3, W L ip sc h itz ia n m app ings f: c lo s u re W w h en ev er x2 + у 2 = 1, and

2 2 2 R n {(x, y): x + у < 1}, c o n s is ts o f a llR3 su ch th a t f(x, y) = ( x ,y ,0 ) E R S

i ( f )"af1 af2 af1 af2"

2+

"af1 9f 3 dî1 9f3]29xJ +

~8f2 af3 af2 af5_Эх Эу Эу Эх. . Эх Эу эу _Эх Эу Эу 9x_

dx dy

14 ALMGREN

fo r f e „ /f . Ф is , o f c o u rse , the tw o -d im e n s io n a l a r e a in te g ra l . Suppose one is g iven the g e o m e tr ic v a r ia t io n a l p ro b le m of finding h e . .^ s u c h th a t Ф(Ь) = in f Ф. The m apping h (x ,y ) = (x, y, 0) i s , of c o u rse , su c h a m app ing . H ow ever, in g e n e ra l, th e re is no r e a s o n fo r a m in im iz in g seq u en ce to ad m it a re a s o n a b le co n v e rg en t su b seq u en ce a s th e follow ing ex a m p le s show:

(1) L e t p j , p 2 , p 3, . . . G R3 be a l i s t o f the p o in ts in R 3 w ith ra tio n a l c o o rd in a te s . One ca n c o n s tru c t a m in im iz in g seq u en ce {f±}¡ w ith lin ij ®(fj)= in f Ф = ir su c h th a t {px, p 2 , . . . , p¡} с f¡(W) fo r each i = 1, 2, 3,fo r exam ple by d efo rm in g th e m ap h above by pu lling th in " te n ta c le s " outof the f la t d isk h(W) in R 3 a s in d ic a te d in F ig . 14, th e to ta l a r e a of w hich " te n ta c le s " is no m o re th an 1 / i . In th is c a s e , ea ch p o in t in R3 is a l im it po in t o f the seq u en ce {f^W )}^

(2) L e t W be g iven the u su a l p o la r co o rd in a te s (r , 6) and define fo r i = 1, 2, 3, . . . th e d iffe o m o rp h ism s

g : W W

g j( r , 0) = ((1 / i )r + (1 - l / i ) r ‘, в)

C lea rly , th e seq u en ce {hogj}¡ is a m in im iz in g seq u en ce fo r the above p ro b le m . H ow ever,

l im ^ h o g j) (x, y) = ( 0 ,0 ,0 ) £ R 3 fo r x2 + y 2 < 1

lim ith o g j) (x, y) = (x ,y , 0) fo r x 2 + y 2 = 1

so th a t no su ita b le co n v e rg en t su b se q u en c es e x is t.

FIG 14. A disk w ith te n ta c le s .

В . 7. G e o m e tr ic in te g ra ls fro m a n o th e r p o in t of view

IA EA -SM R-11/9 15

S uppose Ф: -* R+ is a g e o m e tr ic in te g ra l, F : Rn X G(n, k) -» R+,and fo r each f e

®(f) = J F tp , T an k (f(W ),p)] N (f,p ) d ^ p p e f(W)

Now define f o r ^ kl_f(W) a lm o s t a l l p e R n,

ф{(p) = (p, T a n k(f(W), p ) ) e Rn X G(n, k)

and no te th a t one ca n w rite

®(f) = f Fo<//f d f ^ L N l f , . ) ]Rn

w hich eq u iv a len tly can be w ri tte n

3>(f) = / F d {ф{# [ ^ kL N ( f , . ) l lRn x G(n, k)

in w hich ф^ [gf LN (f, . )] d e te rm in e s a unique R adon m e a s u re o v e r Rn X G(n, k) (assu m in g , say , W is bounded and f is L ip sc h itz ia n ) . Note th a t th is m e a s u re is independen t o f Ф and F . A d e fin itio n is in o rd e r .

B . 8. D efin ition [ 8]

By a k -d im e n s io n a l v a r ifo ld in Rn one m e a n s a R adon m e a s u re o v e r Rn X G (n,'k), Vk (Rn ) deno tes the sp a ce of a ll k -d im e n s io n a l v a r ifo ld s in Rn. In c a s e W is a bounded open s e t in Rk and f: W -» Rn is L ip sc h itz ia n , one d en o tes by

f # |w |e V k (Rn)

the v a r ifo ld co rre sp o n d in g to ¡//f# [^"kL N(f, . )] a s in B . 7. above.

B .9 . R e m a rk

A s w as no ted in B . 7,

®(f) = J f d f#|w | .

One can a lso v e r ify , in a re a so n a b ly s tra ig h tfo rw a rd m a n n e r , th a t fo r each and each d iffe o m o rp h ism g: W -*• W,

f # |w | = (f°g )# IW

16 ALMGREN

so th a t c le a r ly the m apping is no t uniquely d e te rm in e d by i ts a s s o c ia ted v a r ifo ld |w | . O f in te r e s t , how ever, is th e fa c t th a t sp a c e s of R adon m e a s u re s have v e ry s tro n g co n v e rg en ce p r o p e r t ie s in the w eak topology.In th e p r e s e n t con tex t, we have:

B . 10. D efin ition

A seq u en ce {V jj o f R adon m e a s u re s o v e r Rn X G(n, k) ( i .e . a sequence of e le m e n ts of (Rn )) is sa id to converge w eakly to a l im it R adon m e a s u reV if and only if fo r each con tinuous function F : Rn X G(n, k) -* R* w ith co m p ac t su p p o rt,

lim i / F dV¡ = / F dV

B. 11. P ro p o s itio n

A su ffic ie n t cond ition th a t a sequence {V1}i o f R adon m e a s u re s o v e r Rn X G(n, k) co n ta in s a su b seq u en ce {V[.}. w hich c o n v e rg es w eak ly to som e l im it R adon m e a s u re V is th a t fo r eachJ 0 < r < oo,

sup j Vi [B n(0, r) X G(n, k)] < oo

В .1 2 . E x am p les

L e t us r e tu r n to the ex a m p le s of B .6 above w hich i l lu s tr a te d the d iffic u ltie s of th e a tte m p t to use the d ir e c t m ethod in g e o m e tr ic v a r ia tio n a l p ro b le m s . We have c o rre sp o n d in g ex a m p le s .

(1) L e t { fJ i be a s in B . 6(1). Then i t is e a s ily v e r if ie d th a t

lim ¡ f i# IW I = h#|w| (weakly)

w h ere h(x, y ) = (x, у , 0) a s in В . 6, i s a so lu tio n to the p ro b le m in В . 6. G e o m e tr ic a lly , h#[w| c o rre sp o n d s to the unit f la t d isk in R2 X { 0 } c R 3 w hich " s o lv e s" the p ro b le m , but w ithout a p a r t ic u la r p a ra m e tr iz a tio n p r e s c r ib e d fo r th a t d isk .

(2) L e t {g.}. be a s in B . 6(2) above. Then a lso lim j (hog ¡)# | W | =h # |w | (w eakly).Indeed, fo r each i, (hog j)# |w | = h#|w| .

PA R T С . SURFACES AS MEASURES

С. 1. R e aso n s fo r the m e a s u re th e o re tic ap p ro ach

A s has b een su g g e s ted by the v a r io u s ex am p les and d is c u s s io n in P a r t s A and B, one a p p ro a c h to th e s tudy of g e o m e tr ic v a r ia t io n a l p ro b le m s is b a se d on a c o r re sp o n d e n c e betw een su ita b le s u r fa c e s and m e a s u re s on a p p ro p r ia te s p a c e s . Indeed, th e n a tu ra l se ttin g fo r g e o m e tr ic p ro b le m s

IA EA -SM R-11/9 17

in th e ca lc u lu s of v a r ia t io n s s e e m s to be th a t in w hich su r fa c e s a r e re g a rd e d as in tr in s ic a l ly p a r t of Rn (in p a r t ic u la r as m e a s u re s on sp a c e s a s so c ia te d w ith Rn) r a th e r th a n a s m app ings fro m a fixed k -d im e n s io n a l m an ifo ld , even though w ith th is ap p ro a ch one is not ab le to use the t ra d i t io n a l m ethods of fu n ctio n al a n a ly s is fo r show ing the e x is te n c e of so lu tio n s . The m a in re a s o n s fo r doing th is a r e th e follow ing:

(1) M appings f ro m a fixed co m p ac t k -d im e n s io n a l m an ifo ld canno t ta k e in to acco u n t the phenom ena of the ex am p les g iven in P a r t A. F o r ex am p le , one canno t c o n s id e r su r fa c e s of in fin ite to p o lo g ica l type , s u r fa c e s having s in g u la r i t ie s no t r e a liz a b le by m app ings lik e the s in g u la r c u rv e of the t r ip le M obius band, o r su r fa c e s having b o u n d a rie s defined in c e r ta in w a y s .

(2) M any s ig n if ic a n t r e s u l t s have been ob ta ined f ro m the s tudy of g e o m e tr ic v a r ia t io n a l p ro b le m s in the m e a s u re th e o re tic se ttin g in c o n tra s t w ith the v ir tu a l a b sen c e of su ch r e s u l t s in h ig h e r d im e n sio n s and co d im e n sio n s in the m apping se ttin g [1 , 2, 6, 8, 10].

(3) I t s e e m s re a so n a b le to hope th a t once th e s in g u la r i t ie s of the m e a s u re th e o re tic so lu tio n s a r e un d ersto o d , one w ill be ab le to so lv e the m apping p ro b le m as a co n seq u en ce .

(4) T opo log ica l m ethods analogous to th o se of M o rs e 's th e o ry a re a v a ila b le in the m e a s u re th e o re tic se ttin g in c o n tra s t to th e ab sen c e of su ch m e th o d s in the m apping se ttin g [ 11]. F o r exam ple , th e "C ond ition C" of P a la is and S m ale [12] is not s a tis f ie d by any g e o m e tr ic v a r ia t io n a l p ro b le m (note B .6 ) .

(5) The te ch n iq u e s developed in the s tudy of g e o m e tr ic p ro b le m s ina m e a s u r e - th e o re t ic se ttin g have a ided in the so lu tio n of r e la te d p ro b le m s . F o r exam ple :

(a) The b a s ic th e o re m s of c l a s s ic a l in te g ra l g e o m e try have been p ro v e d in w hat s e e m s to be th e ir m o s t n a tu ra l se ttin g [ 13].

(b) V a rio u s long s tan d in g q u es tio n s in th e th e o ry of L eb esg u e a re a have been s e tt le d [ 14].

(c) E x te n s io n s of B e rn s te in 's th e o re m th a t a g lo b a lly defined non- p a r a m e tr ic m in im a l h y p e rsu rfa c e m u s t be a h y p erp lan e have been p ro v ed in d im e n sio n s up to 8 and c o u n te re x a m p le s have been show n to e x is t in h ig h e r d im e n sio n s [ 1 (5. 4. 18), 15] .

(d) P ro o fs have b een g iv en of the r e g u la r i ty a lm o s t ev e ry w h e re of "w eak" so lu tio n s to so m e n o n - lin e a r e llip tic s y s te m s of p a r t ia l d if fe re n tia l eq u a tio n s , and ex am p les exh ib ited w hich show so m e tim e s unique so lu tio n s to su ch s y s te m s w hich co n ta in e s s e n tia l d is c o n tin u itie s [16 , 17].

(6) In the m e a s u r e - th e o re t ic se ttin g , m any im p o r ta n t n a tu ra l g e o m e tr ic c o n s tru c tio n s a r e a v a ila b le , the ana logues of w hich in sp a c e s of m app ings se e m u n n a tu ra l.

C .2 . R e m a rk

T he g e o m e tr ic m e a s u re th e o r i s t ro u tin e ly w o rk s w ith a n u m b er of d if fe re n t m e a s u re s and m e a s u re th e o re tic s u r fa c e s . The follow ing d efin itio n s and te rm in o lo g y a re bo th r e p re s e n ta t iv e of the m o s t com m on " s u r fa c e s " of the g e o m e tr ic m e a s u re th e o r i s t and enab le a c a re fu l s ta te m e n t to be m ad e of so m e o f the r e g u la r i ty and s in g u la r ity r e s u l t s in P a r t s D and E .

18 ALMGREN

C .3 . D efin itio n s and te rm in o lo g y

Suppose 0 s k è n a r e in te g e r s .(1) F o r each A с R n( the k -d im e n s io n a l H au sd o rff m e a s u re of A,

deno ted is the g r e a te s t lo w er bound o f a l l t su ch th a t 0 s t s ooand fo r e v e ry e > 0 th e re e x is ts a coun tab le co v e rin g G o f A w ith

and d iam (S) < e f o r S s G ; h e re a(k) = i / k(Rk n {x: | x | < 1}) [ 1 (2 )].(2) A s u b se t A c R n is ca lle d { ^ k, k) re c tif ia b le if and only if ^И'(А) < oo

and th e re e x is ts fo r ea ch e > 0 a co m p ac t k -d im e n s io n a l d if fe re n tia l su b m an ifo ld M£ of R n of c la s s 1 such th a t ~ M£ ) < e . In c a s e A is £fkm e a s u ra b le one can choose M e so th a t ■$fk([A ~ M e ] и [M e ~ A ]) < e .

(3) In c a s e A с Rn is (gfk, k) re c tif ia b le and m e a s u ra b le ,II Aj| = ^ KL A deno tes th e m e a s u re o v e r R n defined by re q u ir in g ( ^ k L A) (B)= n B) fo r each В с Rn, and T ank (.grkLA, a) d en o tes th e (£?k, k)a p p ro x im a te tan g en t cone to A a t a . [ 1 (3. 2 .1 6 ) ] . F o r k L_ A a lm o s t a ll a in R n, Т ап к(^И*1_ A, a ) e G (n, k). An o r ie n ta tio n fo r A is an gfkLA m e a su ra b le function u: A -» G0(n, k) su ch th a t fo r !gfk a lm o s t a ll a e A, you (a) = T an k0g^kLA, a). H e re G 0(n, k) is the G ra s sm a n n m an ifo ld of a ll o r ie n te d k -d im e n s io n a l p la n es th ro u g h the o r ig in in Rn and y: G0(n, k)-*G(n, k) is th e n a tu ra l p ro je c tio n . The p a i r (A, со) is ca lle d an o r ie n te d (gfk, k) re c tif ia b le and m e a s u ra b le su b se t o f Rn.

(4) If A c R n is (gfk, k) re c tif ia b le and $fk m e a su ra b le , the in te g ra l v a r ifo ld a s so c ia te d w ith A, denoted | A | , is the R adon m e a s u re o v e r Rn defined by

A R adon m e a s u re V e V k (Rn) is ca lle d an in te g ra l v a r ifo ld if and only if it is a f in ite o r co n v e rg en t in fin ite su m of m e a s u re s { |A j|} . co rre sp o n d in g to (gfk, k) r e c tif ia b le and m e a su ra b le s u b se ts А х, A 2, A3 , . . . o f R n . The m o s t a c c e s s ib le s o u rc e fo r th e b a s ic th e o re m s about v a r ifo ld s is R ef. [ 8] to w hich th e r e a d e r s a tten tio n is s tro n g ly d ire c te d .

(5) A k -d im e n s io n a l c u r r e n t in Rn is a continuous l in e a r functional on the sp a ce <fk(Rn ) of a ll d if fe re n tia l fo rm s of d eg re e к and c la s s » on Rn , If к > 0 and T is a k -d im e n s io n a l c u r r e n t in Rn , th en th e boundary of T is a k-1 d im e n s io n a l c u r r e n t defined by se ttin g ЭТ(<р) = T(dip) fo r each ç e & кЛ\ h e re dip is th e e x te r io r d e r iv a tiv e of cp. Suppose the p a i r (A, u) is an o r ie n te d ($?k, k) r e c tif ia b le and m e a s u ra b le s u b se t o f R n and A is bounded. The re c tif ia b le c u r r e n t T a s so c ia te d w ith (A ,u) is g iven by

ss G

[ l Rn X T an k ( ^ kL A , , ) ] # [ ^ k L A ]

re q u ir in g fo r each cp G S k,

p e a

H ere we a r e iden tify ing G0 (n, k) w ith the subm anifo ld of a ll s im p le unit к -v e c to r s in th e G ra s sm a n n v e c to r sp a ce Ak Rn of a ll к -v e c to rs in Rn .

IA EA -SM R-11/9 19

A k -d im e n s io n a l c u r r e n t T in Rn is ca lle d re c tif ia b le if and only if i t is the fin ite o r co n v e rg en t in fin ite su m of r e c tif ia b le c u r r e n ts a s so c ia te d w ith o r ie n te d ( ^ k, k) re c tif ia b le and m e a s u ra b le su b s e ts o f Rn as above. A k -d im e n s io n a l c u r r e n t T is ca lle d a k -d im e n s io n a l in te g ra l c u r r e n t in Rn if and only i f bo th T and ЭТ a r e r e c tif ia b le c u r r e n ts (fo r к = 0, th e re is no re q u ire m e n t re g a rd in g ЭТ w hich, of c o u rse , is not defined).

The m o s t b a s ic th e o re m s about in te g ra l c u r r e n ts in g e n e ra l a r e the d e fo rm a tio n th e o re m [1 ( 4 .2 .9 ) ] , the co m p ac tn ess th e o re m [1 ( 4 .2 .1 7 ) ] , and the a p p ro x im a tio n th e o re m [ 1 ( 4 .2 .2 0 ) ] . The r e a d e r 's a tten tio n is s tro n g ly d ire c te d to th e se fu n d am en ta l fa c ts . The c o m p a c tn e ss th e o re m alone is th e e s s e n t ia l in g re d ie n t in m any e x is te n c e r e s u l t s fo r e l lip tic g e o m e tr ic p ro b le m s by the d ir e c t m ethod . The n e c e s s a ry lo w e r s e m icon tinu ity fo llow s fro m the e llip tic ity . One u su a lly ta k e s as th e "k - d im e n s io n a l a r e a " of a k -d im e n s io n a l in te g ra l c u r r e n t T in Rn the m a s s of T, denoted M (T), defined by se ttin g

M (T) = sup {T(<p): ipeé*1, sup I <p\ s 1}

N ote th a t M (2T) = 2M(T) so th a t, in p a r t ic u la r , the m a s s of T can be m uch l a r g e r th a n ^ f k (sp t T ).

(6) In a n a tu ra l way r e c tif ia b le c u r r e n ts m ay be re g a rd e d as having th e in te g e rs as "c o e ff ic ie n t g ro u p " . F o r each in te g e r v г 2, th e k- d im e n sio n a l f la t ch a in s m odulo v in Rn a r e defined e s s e n tia l ly a s the quo tien t subgroup of the g roup of k -d im e n s io n a l re c tif ia b le c u r r e n ts in Rn induced by the h o m o m o rp h ism Ж -* Ж ¡(2 Ж). T h e o re m s c o rre sp o n d in g to the d e fo rm a tio n th e o re m , th e co m p a c tn e ss th e o re m , and th e ap p ro x im a tio n th e o re m hold fo r f la t ch a in s m odulo v [1 (4. 2. 26)] .

PA R T D. A REGULARITY THEOREM

In th is p a r t we g ive a c a re fu l s ta te m e n t o f a v e ry g e n e ra l r e g u la r i ty r e s u l t fo r so lu tio n s to e l lip tic g e o m e tr ic v a r ia t io n a l p ro b le m s .

D. 1. D efin itio n s

(1) An in te g ra n d F : G(n, к) -» R+ is ca lle d e llip tic w ith e l lip tic ity bound с i f and only if с > 0 and the follow ing cond ition h o ld s. S uppose D с Rn is a f la t k -d im e n s io n a l d isk and S is a co m p ac t \ k) r e c tif ia b le su b se to f R n w hich can not be m apped in to 3D by any L ip sc h itz ia n function f: Rn -* Rn su c h th a t f(p) = p fo r p e S D (it fo llow s th a t 9 D c S ). Then

F(S) - F(D) g с LgÉ^ÍS) - (D)]

(2) An in te g ran d F : Rn X G(n, k) -* R+ is ca lle d e l lip tic if and only if th e re is a con tinuous function c: Rn -> R+ such th a t fo r each p e R N, F p is e l lip tic w ith e l lip tic ity bound c(p); h e re F p: G(n, k) -» R+, F p (7r) = F (p , ir) fo r TreG(n, k).

20 ALMGREN

(3) S uppose 0 Sa < 1 and ej R+ -*■ R+ is m o n o to n ica lly n o n -d e c re a s in gw ith

1

J t ‘(1 + 0° e j t ) 1 dt < oo

о

S uppose a lso F : Rn X G (n, k) -<■ R+ is e llip tic and of c la s s 3. L e t S c R n be k) r e c tif ia b le and B c R n ~ S be c lo sed . One sa y s th a t S is (F ,e J m in im a l w ith r e s p e c t to В if and only if th e re e x is ts 6 > 0 su ch th a t

F (S n { z : <p(z) f z}) = [1 + e =(r)] F (ip [S n{z: <p{ z) f z } ] )

w h en ev er

(a) tp: Rn -> Rn is L ip sc h itz ia n ,

(b) [{ z: cp{z) f z} u cp {z: <p (z) f z} ] л В = ¡3

(c) r = d iam ({z: < p (z ) /z } u < p { z : <p (z) f z i ) < á

D .2 . R e m a rk

In c a se к = n - 1 the e l lip tic ity o f F in D. 1(2) above is eq u iv a len t to the un ifo rm convex ity of F p fo r each p e R n. F o r a ll k, th e s e t o f e l lip tic in te g ra n d s F as in D. 1(1) co n ta in s a convex neighbourhood in the c la s s 2 topology of the к -d im e n s io n a l a r e a in te g ra n d . A lso fo r each d iffeo m o rp h ism f: Rn ->• Rn , f#F is e ll ip tic if and only i f F is , fo r f as in D. 1(2). F in a lly the e l lip tic ity o f F im p lie s th a t th e v a r io u s E u le r -L a g ra n g e equ a tio n s w hich a r i s e a r e n o n - l in e a r s tro n g ly e l lip tic sy s te m s o f p a r t ia l d if fe re n tia l equa tions , and, " in the sm a ll" , the e l lip tic ity o f F is eq u iv a len t to the s tro n g e llip tic ity of th e s e eq u a tio n s .

D. 3. T h eo rem [ 6]

S uppose a , F , S, В a re as in D. 1(3) above and th a t S is (F, e.) m in im a l w ith r e s p e c t to B . Then:

(1) ^ k([S ~ s p t ( ^ kL S )] и [sp tig H 'L S ) ~ (S uB )])= 0

(2) T h e re e x is ts an open su b se t U of R n such tha t:

(a) ^ k[ s p t ( ^ kL.S) - (U u B )] = 0,

(b) s p t ( ^ kl_S )nU is a к -d im e n s io n a l subm an ifo ld of Rn of c la s s 1 w ith ta n g en t к -p la n e s w hich lo c a lly v a ry H o lder con tinuously w ith exponent a .

The s p e c ia l in te r e s t of th is th e o re m is th a t the h y p o th eses a r e su ffic ie n tly w eak to apply to a ll the n a tu ra l c o n s tra in e d g e o m e tr ic v a r ia t io n a l p ro b le m s of w hich I know, the so lu tio n s of w hich y ie ld re c tif ia b le s e t s . The su r fa c e s d is c u s se d in ex a m p le s 2, 5, 6, 7, 8 in P a r t A com e in to th is c a te g o ry in p a r t ic u la r .

IA EA -SM R-11/9 21

V ery l i t t le is known a t the p r e s e n t tim e about the s t r u c tu re of the s in g u la r s e ts of so lu tio n s to g e n e ra l e l lip tic g e o m e tr ic v a r ia t io n a l p ro b le m s , ex cep t fo r th e i r e x is te n c e . H ow ever, fo r the a r e a in te g ran d th e re has been s u b s ta n tia l p r o g r e s s . The follow ing th r e e th e o re m s a r e re p re s e n ta t iv e of the p r e s e n t s ta te of know ledge.

E . l . T h e o re m [10]

L e t В be an n -2 d im e n s io n a l in te g ra l c u r r e n t in Rn su c h th a t ЭВ = 0. Then:

(1) T h e re e x is ts an n -1 d im e n s io n a l in te g ra l c u r r e n t T in Rn such th a t ЭТ = В andM (T) = in f {M(S): S is an n -1 d im e n sio n a l in te g ra l c u r r e n t in Rn and 3S = B}.

(2) T h e re e x is ts an open s e t U in Rn su ch tha t:(a) The H au sd o rff d im e n sio n of sp t T ~ (U и sp t B) is a t m o s t n -8 .(b) sp t T n U is an n -1 d im e n sio n a l r e a l an a ly tic subm anifo ld of Rn

having 0 m e an c u rv a tu re a t each p o in t.

E . 2. T h eo rem [ 10]

L e t l s k l n and le t В be a k -1 d im e n s io n a l f la t cha in m odulo 2 in Rn su ch th a t ЭВ = 0. Then:

(1) T h e re e x is ts a k -d im e n s io n a l f la t ch a in T m odulo 2 in Rn such th a t ЭТ = В andM (T) = inf{M (S): S is a k -d im e n s io n a l f la t ch a in m odulo 2 in Rn and 9S = B}.

(2) T h e re e x is ts an open s e t U in R n su ch th a t:(a) The H au sd o rff d im e n sio n o f sp t T ~ (U.u sp t B) is a t m o s t k -2 .(b) sp t T n U is a k -d im e n s io n a l r e a l an a ly tic subm anifo ld of Rn

having 0 m e an c u rv a tu re a t each po in t.

E .3 . T h eo rem [18]

L e t В be a 1 -d im e n s io n a l f la t ch a in m odulo 3 in R3 su ch th a t ЭВ = 0. Then:

(1) T h e re e x is ts a 2 -d im e n s io n a l f la t cha in T m odulo 3 such th a t ЭТ = В andM(T) = in f {M(S): S is a 2 -d im e n s io n a l f la t cha in m odulo 3 in R3 and dS = B}.

(2) sp t T ~ sp t В = AUC w h ere :(a) A n С = 0.(b) С is a 1 -d im e n s io n a l subm an ifo ld of R 3 of c la s s 1 w ith tan g en t

lin e s w hich lo c a lly v a ry H o ld er con tinuously .(c) A is a 2 - d im e n s io n a l r e a l an a ly tic subm an ifo ld of R3 having 0

m e an c u rv a tu re a t each po in t.(d) E ac h po in t p in С has an open neighbourhood ^ s u c h th a t ^ /n A

h as e x a c tly th re e com p o n en ts , sa y A j, A 2, A 3.F u r th e rm o re fo r ea ch i, A [U (y rn c ) is a H o ld er con tinuously d if fe re n tia b le m an ifo ld w ith boundary , and w h en e v er i f j. A¡ m e e ts Aj along Ж п С a t an ang le of 120°.

PA R T E . ESTIM ATES ON SINGULAR SETS

22 ALMGREN

S ee R ef. [2 ] fo r a co m p le te c la s s if ic a t io n of the in te r io r lo c a l s t r u c tu re of m a th e m a tic a l " so ap bubble" and "so ap f i lm " - l ik e s u r fa c e s , includ ing the f i r s t m a th e m a tic a l v e r if ic a tio n o f the c e n tu ry -o ld "a x io m s of P la te a u " .

E . 4. R e m a rk

R E F E R E N C E S

[ 1] FEDERER, H ., Geometric Measure Theory, Die Grundlehren der m athem atischen Wissenschaften inEinzeldarstellungen, Band 153, Springer-Verlag Berlin - Heidelberg - New York (1969).

[2 ] TAYLOR, Jean E ., Singularities in "soap bubbles" a n d ’’soap film s” , these Proceedings.[3 ] DOUGLAS, J . , Solution of the problem of Plateau, Trans. Amer. M ath. Soc. 33 (1931) 263 -321.[4 ] FLEMING, W .H ., An exam ple in the problem of least area, Proc. Amer. M ath. Soc. 7 (1956) 1065-1074.[5 ] THOM, R ., Quelques propriétés globales des variétés différentiables, Comment. M ath. Helv. 28

(1954) 17 -8 6 .[ 6] ALMGREN, F. J . , J r . , Existence and regularity almost everywhere of solutions to e llip tic variational

problems with constraints, preprint, 294 pages.[7 ] RE1FENBERG, E .R ., Solution of the Plateau problem for m -dim ensional surfaces of varying topological

type (with Appendix by J .F . Adams), Acta M ath. 104 (1960) 1 -9 2 .[8 ] ALLARD, W .K ., On the first variation of a varifold, Ann. of Math. 95 (1972) 417.[9 ] MORREY, C .B ., J r . , M ultiple Integrals in the Calculus of Variations, Die Grundlehren der m athem atischen

Wissenschaften in Einzeldarstellungen, Band 130, Springer-Verlag Berlin — Heidelberg — New York (1966).[ 10] FEDERER, H . , The singular sets of area m inim izing rectifiab le currents with codimension one and of

area m inim izing fla t chains modulo two with arbitrary codimension, Bull. Amer. M ath. Soc. 79(1960) 761 -771 .

[ 11] ALMGREN, F. J . , J r . , The theory of varifolds. A variational calculus in the large for the k-dim ensional area integrand (m ultilithed notes), Princeton (1964).

[12 ] PALAIS, R .S ., SMALE, S ., A generalized Morse theory, Bull. Amer. M ath. Soc. 70 (1964) 165 -172 .[13] BROTHERS, J .E . , Integral geometry in homogeneous spaces, Trans. Amer. M ath. Soc. 124(1966)480-517.[14 ] FEDERER, H . , Currents and area, Trans. Amer. M ath. Soc. 98 (1961) 204 -233 .[ 15] BOMBIER1, E . , DE GIORGI, E . , GIUSTI, E . , M inim al cones and the Bernstein problem, Invent. M ath. 7

(1969) 24 3 -2 6 8 .[16] MORREY, C .B ., J r ., Partial regularity results for non-linear e llip tic systems, J. M ath. M ech. 17

(1968) 649 - 670.[17] GIUSTI, E . , MIRANDA, M . , Sulla rego lariti delle soluzioni deboli di una classe di sistem i e llittic i

quasi-lineari, Arch. Rat. M ech. Anal. 31 (1968/69) 173 -184.[ 18] TAYLOR, J . , Regularity of the singular sets o f 2-dim ensional area m inim izing fla t chains modulo 3

inR 3, Invent. M ath. 22(1943) 119.

IA EA -SM R-11/10

AREA MEASURES ON A REAL VECTOR SPACE

F. BRICKELL Institute o f M athem atics,University of Southampton,United Kingdom

Abstract

AREA MEASURES ON A REAL VECTOR SPACE.In this paper, area measures, transversality and angular m etric are treated . After a discussion of

theorems applying to one-dim ensional measures, there follow sections on scalar products associated with regular area measures as w ell as on area measures on manifolds.

INTRODUCTION

In th is p a p e r , we d is c u s s so m e p ro b le m s in d if fe re n tia l g eo m etry w hich a r i s e in the th e o ry of m -d im e n s io n a l a r e a on a r e a l v e c to r sp a ce of d im e n sio n n (1 S m < n ). The s p e c ia l c a se m = 1 is co n c e rn ed w ith the len g th o r n o rm of a v e c to r and h as been s tud ied in te n s iv e ly . But i t s e e m s lik e ly th a t th e re a re in te re s tin g th e o re m s ye t to be d is c o v e re d in the g e n e ra l c a se .

A p a rt fro m a few r e m a rk s a t the end we have not c o n s id e re d a re a m e a s u r e s on m an ifo ld s . H ere aga in a g re a t d e a l of a tten tio n h as been paid to th e s p e c ia l c a se m = 1 ( F in s le r g eo m etry ).

1. AREA M EASURES, TRANSVERSALITY AND ANGULAR M ETRIC

L e t V be a r e a l v e c to r sp a ce of d im en sio n n w ith s c a la r m u ltip lic a tio n w ritte n on the r ig h t. An m - f ra m e in V is an o rd e re d s e t of m lin e a r ly independen t v e c to rs e ^ e j , , e m]. The s e t of a ll such m - f r a m e s canbe m ade in to a d if fe re n tia b le m a n ifo ld , the S tie fe l m an ifo ld of m - f r a m e s in V. We deno te i t by Vm n. The group L ^ of n o n -s in g u la r m x m m a tr ic e s w ith p o s itiv e d e te rm in a n ts a c ts on Vm n on th e r ig h t ac co rd in g to the ru le

e = [e 1 ; ------e m] -[e1, --------i m] = еф

w here

e 8 = У| еа^Вa

and ф i s the m a tr ix [0 “ ], T he quo tien t sp a ce is the G ra ssm a n n m an ifo ld of o r ie n te d m -p la n e s in V. We deno te i t by G min.

0 23

24 BRICKELL

An a re a m e a s u re of d im en sio n m on V is a p o s itiv e d iffe re n tia b le function L on Vm n w hich s a t is f ie s the hom ogeneity cond ition ,

E x am p le . An a re a m e a s u re of d im en sio n 1 is a p o s itiv e hom ogeneous function L of d e g re e 1 defined on Vj n = V - 0.

E x am p le . A s c a la r p ro d u c t on V defines an a re a m e a s u re of d im en sio n m fo r each m , 1 S m < n . The m e a s u re is d e te rm in e d un iquely (and co n s is te n tly ) by the cond ition L(e) = 1 fo r any o r th o n o rm a l m - fra m e e. An a re a m e a s u re L is s y m m e tr ic if Ъ(еф) = ( |d e t 0 ) ) L(e) fo r any n o n -s in g u la r m x m m a tr ix ф.F o r ex am p le , the a r e a m e a s u re s d e te rm in e d by a s c a la r p ro d u c t a re a ll sy m m e tr ic .

We sh a ll often m ake u se of a b a s is [ E j , . . . , E n] of V. G iven such a b a s is the v e c to rs of an m - f ra m e e = [e 1, . . . , e m] can be e x p re s se d uniquely as

and the functions pj, d e te rm in e a g lobal c h a r t on Vm n . We w rite f f= 3 f /3 p ^ fo r any r e a l valued d if fe re n tia b le function f on Vm n . W ith th is no ta tion it can be show n th a t the hom ogeneity condition (1 .1 ) im p lie s th a t

w h ere 6g = 0 (af¡3), б “ = 1.C o n s id e r a p lane 3rGGm n and choose an m - fra m e e in ir. A v e c to r

J e . V is t r a n s v e r s a l to w if 2 L “(e) X1 = 0, a = 1, . . . , m . It can be i i

shown th a t th is defin itio n is independen t of the cho ice of e in n and of the b a s is [ E j , . . . , E n] . T he s e t of a l l v e c to rs t r a n s v e r s a l to v fo rm an (n -m )-p la n e (n o n -o rie n ted ) ca lle d the t r a n s v e r s a l p lane to v.

The an g u la r m e tr ic is a q u a d ra tic d if fe re n tia l fo rm on Gm n w hich we define in the fo llow ing w ay. P u t

Ы е Ф) = (d e t (//) L (e), ф £ Ь +т ( 1 . 1 )

( 1. 2)

. . , m

and c o n s id e r th e q u a d ra tic d if fe re n tia l fo rm on Vm n

It can be show n th a t H is independen t of the b a s is [E 1, . . . , E n]. F u r th e r , i f ш: Vm n G mjI1 is the n a tu ra l p ro je c tio n then H is the im ag e u n d e r ¡3* of a q u a d ra tic d if fe re n tia l fo rm on G m_n . We c a ll th is fo rm the an g u la r m e tr ic .

IAEA-SM R-11 /10 25

E x am p le . Suppose th a t m = 1, n = 2 and in tro d u c e p o la r c o -o rd in a te s r , ti on Vi 2 =V - 0. We also r e g a rd в as a lo c a l c o -o rd in a te on G li2, the m a n ifold of o r ie n te d 1 -p lan e s o r d ire c tio n s in V. The a re a m e a s u re L can be e x p re s s e d a s L = rf(0 ) w here f is a p o s itiv e d if fe re n tia b le function of Э, and a c a lc u la tio n show s th a t the an g u la r m e tr ic is

( l + (f" / f ) ) d 6*2

It is n a tu ra l to ask the ex ten t to w hich t r a n s v e r s a l i ty o r the an g u la r m e tr ic d e te rm in e s the a re a m e a s u re . It is v e ry ea sy to p ro v e by lo c a l a rg u m e n ts th a t L and L d e te rm in e the sa m e t r a n s v e r s a l p la n es if , and only if , L is a co n s tan t m u ltip le of L . On the o th e r hand , th e re a re v a lu es of m ,n fo r w hich the p ro o f of the fo llow ing th e o re m needs a g lobal a rg u m en t [1 -3 ].

T h e o re m 1. Two a re a m e a s u r e s , L , L d e te rm in e th e sa m e an g u la r m e tr icif , and only if , L i s a co n s tan t m u ltip le of L.

P ro o f . The su ffic ien cy of the cond ition is obv ious. We w ill p rove its n e c e s s i ty fo r a r e a m e a s u re of d im en sio n 1 (the m ethod ex tends to the g e n e ra l ca se ) . We define a function a on V - 0 by L = aL . The hom ogeneity cond ition on L , L im p lie s th a t a =ct ° w w h ere a i s a d if fe re n tia b le function on the co m p ac t sp a ce Gj n . C onsequen tly a a tta in s a m ax im u m v alue in V - 0.

On th e o th e r hand , s in c e L , L d e te rm in e the sam e an g u la r m e tr ic , a s a t is f ie s the p a r t ia l d if fe re n tia l equa tions

Э2а 1 ida 9L 9a 3 L i . .9p¡ 3pJ Ь 'Э р ‘ 9рГ 9pi 9 p ï '“ J l , J -------- П

w h ere we have w ritte n p1 fo r . It follow s th a t a s a t is f ie s the e llip tic equation

V 92 q 2 V 9j 9LL (Эр1)2 L L Эр1 9pii i

and th e re fo re th e m ax im u m p r in c ip le of Hopf [4, p. 61] im p lie s th a t a isa c o n s ta n t function . We r e m a rk th a t the p re v io u s exam ple show s th a t ag lobal a rg u m e n t is c e r ta in ly n e c e s s a ry in the c a se m = 1, n = 2. F o r if f is any l in e a r function then th e lo c a lly defined m e a s u re L = rf(0 ) h as the sam e an g u la r m e tr ic a s L = r .

An a r e a m e a s u re i s sa id to b e r e g u la r if i t s an g u la r m e tr ic is p o s itiv e d e fin ite . An eq u iv a len t cond ition is th a t H be p o sitiv e s e m i-d e f in ite of ra n k m (n -m ).In th e c a se m = 1 a s y m m e tr ic r e g u la r a r e a m e a s u re is a n o rm on V. F ro m now on, we sh a ll be c o n c e rn ed only w ith r e g u la r a re a m e a s u re s .

A n o th er n a tu ra l q u es tio n of a g e n e ra l n a tu re is w h e th e r th e re is a d ua lity betw een a r e a m e a s u re s of d im e n sio n s m and n -m . To a n sw e r th is we in t r o duce the id e a of a c o n fo rm a i a re a m e a s u re . We say th a t two a re a m e a s u re s of the sa m e d im en sio n m a re co n fo rm a i if they a re c o n s ta n t m u ltip le s of each o th e r . T h is eq u iv a len ce re la t io n p a r t i t io n s the a r e a m e a s u re s into

26 BRICKELL

eq u iv a len ce c la s s e s w hich a r e c a lle d c o n fo rm a i a re a m e a s u re s of d im en sio n m . It is c le a r tha the d e fin itio n s of t r a n s v e r s a l i ty and an g u la r m e tr ic a re v a lid fo r a co n fo rm a i a re a m e a s u re .

We r e s t r i c t o u rs e lv e s to s y m m e tr ic m e a s u re s so th a t t r a n s v e r s a l i ty can be re g a rd e d a s a function betw een G ra ssm a n n m an ifo ld s of n o n -o rie n te d p la n e s . A lso the an g u la r m e tr ic s can be re g a rd e d as R iem ann ian m e tr ic s on th e se m a n ifo ld s . The p ro o f of the follow ing th e o re m is con tained in R ef. [5].

T h eo rem 2 . L e t ¡/’i, ¿ V i denote the s e ts of s y m m e tr ic co n fo rm a i a re a m e a s u re s on V of d im e n sio n s 1 and n -1 re s p e c tiv e ly . Then th e re is a b ije c tio n *-» such th a t the c o rre sp o n d in g t r a n s v e r s a l i ty functions a re in v e r s e s of each o th e r and a re i s o m e tr ie s of the c o r re sp o n d in g an g u la r m e tr ic s .

P re su m a b ly , i t would not be d ifficu lt to ex tend th is th e o re m to m e a s u re s of a r b i t r a r y d im en sio n .

2. ONE-DIM ENSIONAL MEASURES

In th is p a ra g ra p h we d is c u s s two th e o re m s w hich apply in the s p e c ia l c a se m = 1. F o r b rev ity we c a ll a s y m m e tr ic a re a m e a s u re of d im en sio n 1 a d if fe re n tia b le n o rm . Note a lso th a t G j n, the m an ifo ld of d ire c tio n s in V, is d iffeo m o rp h ic to th e s ta n d a rd (n - l) - s p f ie re .

L et u s sup p o se th a t n > 2 . A c a lc u la tio n show s th a t if the n o rm L is defined by a s c a la r p ro d u c t on V then i t s an g u la r m e tr ic h as co n s tan t c u rv a tu re eq u a l to 1. C o n v e r s e ^ we h a v e ^ - S ] ,

T h eo rem 3 . Suppose th a t L is a d if fe re n tia b le n o rm on a r e a l v e c to r sp a ce V of d im en sio n > 2 , and th a t i ts an g u la r m e tr ic has co n s tan t c u rv a tu re equal to 1. Then L is d e r iv e d f ro m a s c a la r p ro d u c t on V.

We m ake so m e r e m a rk s about the p ro o f of th is th e o re m . L e t u s choose a s c a la r p ro d u c t on V and deno te th e c o rre sp o n d in g n o rm by L . It follow s fro m (R ef. [9], C h a p te r 6, T h eo rem 7^_10) th a t th e re is an is o m e try Gj n->- Gj n betw een the an g u la r m e tr ic s of L and L. T h eo rem 3 can be p ro v ed by show ing th a t the is o m e try is the p ro je c tio n of a l in e a r tr a n s fo rm a tio n V - 0 -» V - 0, and th is is e s s e n tia l ly w hat is done in R ef. [6].

The m ethod r a i s e s the fo llow ing question w hich m a k es se n se fo r m e a s u re s of any d im e n sio n . F i r s t note th a t any n o n -s in g u la r l in e a r tra n s fo rm a tio n V -» V in d u c e s , in an obvious w ay, a d if fe o m o rp h ism Gm in-> Gm_n . Now suppose th a t f: G m n -> Gm_n is an is o m e try betw een the an g u la r m e tr ic s of two a r e a m e a s u r e s of the sa m e d im e n sio n . Is f n e c e s s a r i ly induced by a l in e a r tr a n s fo rm a tio n V-*V?

The second th e o re m we w ish to m en tion is due to S ch n eid er [7 ].

T h eo rem 4. Suppose th a t L is a d if fe re n tia b le n o rm on a r e a l v e c to r sp a ce of d im en sio n n and le t WL be th e vo lum e of Gj n in th e an g u la r m e tr ic of L. Then

w here С,,.! is the vo lum e o f the u n it ( n - 1 ) - s p h e re , and the equa lity ho lds if and only if L is d e riv e d fro m a s c a la r p ro d u c t on V.

IA EA -SM R-11/10 27

T h e o re m 3 can be deduced e a s ily f ro m T h eo rem 4. F o r if the an g u la r m e tr ic o f L h as co n s tan t c u rv a tu re 1 then G1>n w ith th is m e tr ic is is o m e tr ic w ith the u n it ( n - l ) - s p h e r e (R ef. [9], C h a p te r 6, T h eo rem 7. 10) and, conse q u en tly , WL = Cn_ i.

We do not know if the an g u la r m e tr ic defined by a s c a la r p ro d u c t h as a c o r re sp o n d in g e x tre m a l p ro p e r ty in d im en sio n m > l . A no ther q u es tio n is w h e th e r the sy m m e try cond ition im p o sed on L in T h eo rem 3 is n e c e s s a ry .It is known th a t i t can be o m itted p rov ided th a t V h as d im en sio n 3 and L is an a ly tic [8].

T h e o re m s analogous to T h e o re m s 3 and 4 a re t r u e fo r m e a s u re s of co d im en s io n 1. They can , e . g. be deduced by m e an s of T h eo rem 2.

3. SCALAR PRODUCTS ASSOCIATED WITH REGULAR AREA MEASURES

L e t S(V) denote the m an ifo ld of a l l s c a la r p ro d u c ts on V. A d if fe re n tia b le function Gm n -*■ S(V) w ill be ca lle d an S -m a p . G iven a r e g u la r a r e a m e a s u re L of d im en sio n 1 we can define an S -m ap G i n S(V) in the follow ing way. C hoose a b a s is [E 1; . . . , E n ] of V, w rite p 1 fo r p* and put

, , 92( iL 2), . . ,^ (*> = э 1*3 = 1 ...........n

w h ere e is any n o n -z e ro v e c to r in тг. The m a tr ix [g ij( тг)] can be shown to be p o s itiv e d e fin ite and th e re fo re a un ique s c a la r p ro d u c t <( , )> is d e te r m in ed by the cond itions

< E j, E j > = g ij(7r)

The s c a la r p ro d u c t is independen t of the chosen b a s is .We pau se to m en tio n the fo llow ing th e o re m due to D eicke [10 ,1 1 ].

T h eo rem 5. Suppose th a t the d e te rm in a n t of th e m a tr ix [g¡j ] is co n s tan t on GitD. Then the m a tr ix [gy ] is c o n s ta n t and th e re fo re the m e a s u re L is d e r iv e d fro m a s c a la r p ro d u c t on V.

An S -m ap Gn- i>n S(V) can be defined in the c a se of co d im en sio n 1, again by a s im p le fo rm u la [12]. T h e o re m 5 is s t i l l v a lid in th is c a se .B ecau se of the fu n d am en ta l im p o rta n c e of th e se id e a s in the w ork of C a rta n [12 ,13 ] i t is n a tu ra l to t r y and define an S -m ap fo r an a re a m e a s u re of a r b i t r a r y d im e n sio n . It is e s s e n t ia l th a t the defin itio n be in t r in s ic , th a t i s , an au to m o rp h ism of V w hich t r a n s f o rm s a m e a s u re L in to a m e a s u re L m u st a lso t r a n s fo rm th e c o rre sp o n d in g S -m a p s in to each o th e r . We w ill d e s c r ib e one way of d ea lin g w ith th is p ro b lem .

L e t G deno te the g roup S L mx L n_m , so th a t an e lem e n t of G is an o rd e re d p a i r ( r , s ) w h ere r is an m x m m a tr ix w ith d e te rm in a n t 1 and s is a n o n -s in g u la r (n -m )x (n - m ) m a tr ix . L e t 38 denote the m an ifo ld of p o s itiv e d e fin ite n X n m a tr ic e s of the fo rm

h 0

0 к

28 BRICKELL

w h ere h i s of o r d e r m x m and h as d e te rm in a n t 1. G a c ts on 33 on the r ig h t by

'h 0 1 r Th r 0

_0 к J 0 sTks _

P u t N = m (n -m ) and le t denote the m an ifo ld of p o s itiv e d efin ite NxN m a tr ic e s . We sh a ll w rite

X = [X“b6]

fo r a ty p ic a l e le m e n t, the row s and co lum ns of X being indexed by the p a irs

, or, ,8 = 1 . . . . , m ; a , b = m + 1, . . . , n. G ac ts on & on the r ig h t by

X ^ X^r?r,® £

X , 11, C , d

,-1Suppose now th a t L is a r e g u la r a r e a m e a s u re of d im en sio n m . We

fix a p lane TrGGm n and c o n s id e r th o se n - f r a m e s b = [b j , . . . , b n] of V suchth a t b 1; b m a re in 7Г, L (b i , bm) =1 and bm + 1 1 bn a re in the(n -m )-p la n e t r a n s v e r s a l to w. The group G a c ts on the r ig h t on th e union of th e se f ra m e s and, a s n v a r ie s , we get a p r in c ip a l bundle В o v e r Gm_n w ith group G.

Suppose th a t x: В SB is an e q u iv a r ia n t m ap , th a t i s x(bg) = x(b)g fo r g e G . T hen, g iven 7reGm n , we can d e te rm in e a s c a la r p ro d u ct on V by the condition

[< b¡, b ^ ] = x(b)

w h ere b is a f ra m e in the f ib re o v e r 7r. B ecau se x is e q u iv a r ia n t the s c a la r p ro d u c t i s independen t of the cho ice of b. It h as the p ro p e r tie s

(i) the m e a s u re of an m - f ra m e in n i s the sa m e as th a t defined by the s c a la r p ro d u c t;

(ii) a v e c to r w hich is t r a n s v e r s a l to л i s o rth o g o n a l to тт.C o n v e rse ly any S -m ap w ith th e se p ro p e r t ie s can be d e r iv e d fro m an equ iv a r ia n t m ap В ->38.

To c o n s tru c t such an e q u iv a r ia n t m ap , we sh a ll m ake u se of the an g u la r m e tr ic . T h e re is a n a tu ra l m ap X of the bundle В in to the p r in c ip a l bundleof tan g en t f ra m e s to G r It can be defined as fo llow s. C o n s id e r a f ra m eb e В and deno te i t s p ro je c tio n in Gm_n by тт. T hose m -p la n e s con ta in ing a (n e c e s s a r i ly unique) m - f ra m e of v e c to rs

b„ + I < b a , a = 1 , , m ; a = m + l , . . . , n

d e te rm in e a c o -o rd in a te neighbourhood of 7r w ith c o -o rd in a te functions u^. The f ra m e X(b) i s the m (n -m )- f ra m e f of v e c to rs f “ w here

IA EA -SM R-11/10 29

We put С = A(B). The m ap X i s e q u iv a r ia n t w ith r e s p e c t to the ac tion of G on С defined by

f = [ f ? ] - [ X f№а,ь

The an g u la r m e tr ic , w hich is a R iem ann ian m e tr ic on Gm n , en a b le s us to define a fu r th e r e q u iv a r ia n t m ap p: by

f - [ x S l

w h ere = < fa< f b ^ F in a lly le t г: á®-» SB be a d if fe re n tia b le e q u iv a r ia n t m ap . The co m p o sitio n r » p » X: B -> ^ i s e q u iv a r ia n t and can be u se d to define an S -m ap in tr in s ic a lly a s so c ia te d w ith the r e g u la r a r e a m e a s u re L.

Of c o u r s e , we s t i l l have to p ro v e th a t such e q u iv a r ia n t m ap s r : &>-* SB e x is t. R ef. [14] co n ta in s an ex is te n c e p ro o f fo r a g e n e ra l v a lue of m but the m ethod c e r ta in ly does not le ad to any e x p lic it a lg e b ra ic fo rm u la . It is not d ifficu lt to se e why th e re a re s im p le fo rm u lae fo r S -m a p s in the s p e c ia l c a s e s m = 1 and m = n - 1. F o r in th e se c a s e s G a c ts tr a n s i t iv e ly on and th e re a r e unique e q u iv a r ia n t m a p s w hich tak e the un it m a tr ix in to the u n it m a tr ix i n ^ .

One way of study ing a re a m e a s u re s is th rough the o rb it s t r u c tu re of p(C). If m = 1 o r n - 1 then p(C) is the o rb it con ta in ing the un it m a tr ix in Sfi and it is n a tu ra l to ask fo r the o th e r m e a s u re s w hich have th is p ro p e r ty .The a n sw e r, due e s se n tia lly to T andai [1 5 ], is th a t they a re ju s t the m e a s u re s d e r iv e d fro m a s c a la r p ro d u c t on V. We w ill lead up to a p ro o f o f th is fac t by c o n s id e r in g lo c a l se c tio n s of the bundle B. We r e m a rk th a t R ef. [16] co n ta in s a m o re d e ta ile d study of th is bundle.

L et b = [ b j , . . . , b n] be a lo c a l se c tio n of B. C hoosing a fixed b a s is [ Е 1; . . . , E n] fo r V we can e x p re s s b in te rm s of th is b a s is as

i , j = 1, . . . , nÍ

We w rite Q fo r the m a tr ix [qj ] w h ere i and j index th e row s and co lu m n s, r e s p e c tiv e ly , and in tro d u c e the m a tr ix o f d if fe re n tia l fo rm s П = Q '1dQ.The m a tr ix fi is independen t of the chosen b a s is fo r V and s a t is f ie s the re la tio n

dQ + Q A fi = 0 (3. 1)

w h ere л d en o tes the e x te r io r p ro d u c t. The fo llow ing c a lc u la tio n s r e la te П to the a re a m e a s u re .

L e t [jujlj] be the dual b a s is to X(b). A s h o r t c a lc u la tio n show s th a t

^ q? dqjj, <2 = 1 , . . . , m ; a = m + l , . . . , nj

w h ere qj1 a r e e le m e n ts of the in v e rs e m a tr ix Q"1 = [qj ] . The an g u la r m e tr ic can , of c o u rs e , be e x p re s se d in te rm s of the d if fe re n tia l fo rm s jj“ .

30 BR1CKELL

To do th is we need the id e n tity

pj, = 0 (3.2)J

w hich can be d e r iv e d fro m the id en tity (1. 2). We w rite

d q j * = X 61 d q “ = Z ( Z + Z 4 e ^ ) d q “ = ^ ^ a + ^ e ^ f d q aj j а 8 a j. В

It fo llow s by u s in g the id e n tity (3. 2) th a t the an g u la r m e tr ic

£ H íf(e )d q ¿ d q ]s = £ Х “® м У 6

a.6,i,j ot,B, a,b

w h ere

K b = X H e)qa4b i.i

and e d en o tes the m - f ra m e [ b i , . . . , bm].The id e n tity (1.2) and th e t r a n s v e r s a l i ty condition on the f ra m e s in В

to g e th e r im ply th a t

— a 1 T a, *q t = r L i ( e )

C o nsequen tly , a c a lc u la tio n show s th a t

^ q f d q à = - ^ q ' a d q f = - ^ Xfbn\ i i 8,b

F in a lly , i t follow s fro m the cond ition L(e) =1 th a t

^ q “d q « = о

The p re c e d in g c a lc u la tio n s have the follow ing co n seq u e n ces . The m a tr ix

S3 =в v

Ф

w h ere ц i s the m a tr ix [ /4 J and в, ф, v a r e m a tr ic e s of d if fe re n tia l fo rm s w ith t r a c e 0 = 0. The an g u la r m e tr ic is - t r a c e (/1• v) w h ere ц • v i s the s y m m e tr ic p ro d u c t. We u se th e se id e a s and n o ta tio n s to p rove

T h eo rem 6. If the a re a m e a s u re is d e r iv e d fro m a s c a la r p ro d u c t on V then p(C) is the o rb it con ta in ing the u n it m a tr ix in The c o n v e rse s ta te m e n t is t r u e p ro v id ed th a t 2 S m S n - 2.

IAEA-SM R-11/10 31

P ro o f. Suppose th a t the a re a m e a s u re i s defined by a s c a la r p ro d u c t on V. We choose the lo c a l se c tio n b of В and the fixed b a s is [ E j , . . . , E n] to be o r th o n o rm a l. C o nsequen tly , the m a tr ix Q is o r th o g o n a l, Q is skew -sy m m e tr ic and th e re fo re the an g u la r m e tr ic is 2 ^ follow s th a t

ot, athe b a s is f = A.(b) is o r th o n o rm a l and th a t p(f) is the u n it m a tr ix in T h e re fo re , p(C) is the o rb it of £0 w hich co n ta in s the un it m a tr ix .

C o n v e rse ly if p(C) is th is o rb it then th e re is a unique e q u iv a r ia n t m ap t : p(C)-+3§ w hich ta k e s the un it m a tr ix of ¿0to the un it m a tr ix of SS. We u se the co m p o sitio n т ° p ° À to define an S -m ap Л: Gmtn ^ S(V), and we sh a ll co m p le te the p ro o f by show ing th a t if 2 S m S n - 2 then A is c o n s tan t.

To do th is , we choose the lo c a l se c tio n b so th a t Ь(гг) is o r th o n o rm a l w ith r e s p e c t to the s c a la r p ro d u c t A(ir). It follow s th a t th e an g u la r m e tr ic i s 2 ( ^ â ) 2 and th e re fo re

а, а

L e t [ E j , . . . , E n] be a fixed b a s is fo r V and denote the m a tr ix of s c a la r p ro d u c ts K E ¡ , E j )>] by K. A c a lc u la tio n show s th a t

dK = - (Q '1 )T ( Г2 + f2T) Q '1

and, co n seq u en tly , we need to p ro v e th a t Q is s k e w -sy m m e tr ic .I t fo llow s fro m re la t io n (3. 1) th a t

d/u + /uA6+0A/u = O, d;uT + 9 Л д T + д т A 0 = 0

We tr a n s p o s e the second equa lity and s u b tr a c t i t f ro m th e f i r s t to ob ta in

M A (0 + 0 T) + (ф + ) А ц = 0

We o m it the e le m e n ta ry a lg e b ra w hich show s th a t if 2 i m á n - 2 then th is re la t io n , to g e th e r w ith the fac t th a t в + в т h as z e ro t r a c e , im p lie s tha t 0+0^ =0, ф + ф^=0.

4. AREA MEASURES ON MANIFOLDS

A re g u la r a r e a m e a s u re on a d if fe re n tia b le m an ifo ld M of d im en sio n n is a d if fe re n tia b le a s s ig n m e n t of a r e g u la r a r e a m e a s u re to ea ch tangen t sp a ce Tp(M ), p £ M . We sup p o se th a t the m e a s u r e s have d im en sio n m and deno te the m an ifo ld of o r ie n te d tan g en t m -p la n e s to M by ¡Й. L et a: M-»M be the n a tu ra l p ro je c tio n . The w ork in se c tio n 3 im p lie s th a t th e re e x is ts a s c a la r p ro d u c t in the induced v e c to r bundle a ‘1(TM) w hich is in tr in s ic a lly a s so c ia te d w ith the a r e a m e a s u re .

An im p o r ta n t co n seq u en ce is th a t we can a ttem p t to c o n s tru c t a connection in c r^ T M ) by a p ro c e d u re ana logous to th a t u sed to c o n s tru c t the R iem ann ian connection in R iem an n ian g eo m etry . The p ro c e d u re i s s u c c e s s fu l in the c a s e m = 1 [13] and le a d s to a so lu tio n of the lo c a l eq u iv a len ce p ro b lem .

32 BRIC KELL

F o r m > l the p seu d o -g ro u p of lo c a l a u to m o rp h ism s can be in fin ite d im e n sio n a l and i t is n e c e s s a ry to im p o se r e s t r ic t io n s on the a re a m e a s u re . We r e f e r to R ef. [12] fo r d e ta ils of the c a se m = n - 1.

In R efs [14, 3] the s c a la r p ro d u c t is u sed in a d if fe re n t way to p rov ide a n s w e rs to so m e s im p le g lobal q u es tio n s about a re a m e a s u re s on a m anifo ld .

The fo rm u la fo r the s c a la r p ro d u c t in the c a se m = 1 h as been g e n e ra liz e d in a d if fe re n t d ire c tio n by Iw am oto [17]. F o r a r e g u la r a r e a m e a s u re of a r b i t r a r y d im en sio n m he h as given an ex p lic it fo rm u la w hich a s s o c ia te s w ith each 7reGm n a s c a la r p ro d u c t on the m - v e c to r s of V. We r e f e r to R efs [18 ,19 ] fo r an ap p ro ach to th e th e o ry of a r e a m e a s u re s w hich m akes u se of Iw a m o to 's w ork .

R E F E R E N C E S

[1] DELENS, P., La métrique angulaire des espaces de Finsler, Actualités Sci. et Ind. No. 80 Paris (1934).[2] GOLAB, S., Sur la représentation conforme de deux espaces de Finsler, C.R. Acad. Sci. Paris 196

(1933) 986.[3] BRICKELL, F., AL-BORNEY, M.S., A note on area measures, J. London Math. Soc. 4 (1972) 466.[4] MORREY, C.B., Multiple Integrals in the Calculus of Variations, Springer-Verlag, N e w York (1966) 61.[5] BRICKELL, F., A relation between Finsler and Cartan structures, Tensor, N.S. 2J3 (1972) 360.[6] BRICKELL, F., A theorem on homogeneous functions, J. London Math. Soc. 42 (1967) 325.[7] SCHNEIDER, R., Über die Finslerrâume mit Sjj^^ 0t Arch. Math. 19 (1968) 656.[ 8] MÜNZNER, H. F., Die Poincarésche Indexmethode und ihre Anwendungen in der affinen Flachentheorie,

Dissertation FU Berlin (1963).[9] KOBAYASHI, S., NOMIZU, K . , Foundations of Differential Geometry, _1, N e w York and London (1963).

[10] DEICKE, A., liber die Finslerrâume mit A¿=0, Arch. Math. 4 (1953 ) 45.[11] BRICKELL, F., A new proof of Deicke's theorem on homogeneous functions, Proc. Amer. Math. Soc. 1£

(1965) 190.[12] CARTAN, E., Les espaces métriques fondés sur la notion d’aire, Actualités Sci. Ind. No. 72, Paris (1933).[13] CARTAN, E., Les espaces de Finsler, Actualités Sci. Ind. No. 79, Paris (1934).[14] BRICKELL, F., Differentiable manifolds with an area measure, Can. J. Math. Г9 (1967) 540.[ 15] TANDAI, K . , On areal spaces VI. Tensor (N. S. ) 3 (1953) 40.[16] WAGNER, V . , Geometry of a space with an areal metric and its applications to the calculus of variations,

Rec. Math. 19 (1946) 341 (in Russian).[17] I W A M O T O , H . , On geometries associated with multiple integrals, Mathematica Japónica ¿(1948) 74.[18] DAVIES, E. T. , Areal spaces, Ann. Mat. Pura Appl. 4 55 (1961) 63.[19] K A W A G U C H I , A., On the theory of areal spaces, Bull. Calcutta Math. Soc. 56 (1964) 91.

IA EA -SM R-11/11

THE DIFFERENTIABILITY OF TRANSFORMATIONS WHICH PRESERVE GEODESICS

F. BRICKELL Institute o f M athem atics,University o f Southampton,United Kingdom

Abstract

T H E DIFFERENTIABILITY OF TRANSFORMATIONS W H I C H PRESERVE GEODESICS.The aim of this paper is to study under which extra conditions a bijection f of a smooth Riemannian

manifold M which preserves geodesics is a diffeomorphism. The theorem given here (with proof) is: If f is a homeomorphism and dim M > 2 then f is a diffeomorphism. The arguments involve only basic ideas in differential geometry.

T h is co n trib u tio n is b ased on a p a p e r pub lished som e tim e ago in P ro c . Am. M ath. Soc. J_6 (1965) 567-574. T he top ic is r e a so n a b ly g e n e ra l and th e a rg u m e n ts involve only b a s ic id e a s in d if fe re n tia l g e o m e try . In ad d itio n , i t s e e m s lik e ly th a t the r e s u l t could be im p ro v e d . We sh a ll beg in by d e s c r ib in g two th e o re m s in each of w hich a s im p le g e o m e tr ic a l cond ition on a tr a n s fo rm a tio n is shown to have a s u rp r is in g ly s tro n g co n seq u en ce .

L e t Vn deno te a r e a l v e c to r sp ace of d im en sio n n, and sa y th a t nonz e ro v e c to rs u , v e V n a r e eq u iv a len t if u = Xv fo r som e n o n -z e ro r e a l n u m b e r X. An eq u iv a len ce c la s s is ca lle d a n o n -o rie n te d d ire c tio n in Vn bu t in th is co n trib u tio n we sh a ll o m it the w ord n o n -o rie n te d . The se t P n_1 of a l l d ire c tio n s in Vn is ca lle d r e a l p ro je c tiv e sp a ce of d im en sio n n -1 . A su b se t of d ire c tio n s co rre sp o n d in g to the n o n -z e ro v e c to rs in a tw o -d im e n s io n a l su b sp a ce of Vn is ca lle d a line in p n_1 .^ A n o n -s in g u la r l in e a r tr a n s fó rm a tio n ф: Vn -*■ Vn in d u ces a b ije c tio n ф: p n_1 -» p n_1 w hich ta k e s lin e s in to lin e s . S uppose, c o n v e rse ly , th a t f: P n 1 -* P n 1 is a b ije c tio n w hich ta k e s lin e s in to lin e s . Is f induced by a n o n -s in g u la r l in e a r tr a n s fo rm a tio n of Vn ? Of c o u rse , th is is not so if n = 2 fo r then P n_1 = P 1 c o n s is ts of ju s t one line and any b ije c tio n p r e s e rv e s th is lin e . H ow ever, fo r n > 2 it can be p roved th a t f is induced by such a t ra n s fo rm a tio n , and th is fac t is known a s the fundam en ta l th e o re m of r e a l p ro je c tiv e g eo m etry .

T he second th e o re m w hich we w ish to m en tion is a th e o re m in R iem an n ian g e o m e try . L et M be a C“ R iem an n ian m an ifo ld . Then M can be m ade in to a m e tr ic sp a ce and we denote the d is ta n c e function by p. A b ije c tio n f: M -» M is sa id to be an is o m e try if p(f(x), f(y)) = p(x, y) fo r a l l x , y e M . The th e o re m of M y e rs -S te e n ro d s ta te s th a t an is o m e try of a R iem an n ian m an ifo ld is n e c e s s a r i ly a d iffe o m o rp h ism .

In w hat fo llow s we s h a ll a lso be w ork ing w ith a C” R iem an n ian m anifo ld M and it w ill be im p o r ta n t to b e a r in m ind the d is tin c tio n betw een a g eo d esic cu rv e and i ts ra n g e . We sh a ll u se the w ord g eo d e sic to m ean the ran g e

33

34 BRICKELL

of a g eo d e sic c u rv e . F u r th e rm o re , a b ije c tio n f: M -* M w ill be sa id to p r e s e rv e the g e o d e s ic s of M if, fo r any m a x im a l g eo d esic gC M , both f(g) and f -1(g) a r e a lso m a x im a l g e o d e s ic s of M.

H aving e s ta b lish e d th is n o ta tio n we can s ta te the m ain a im of th is p a p e r . It is to c o n s id e r the follow ing q u estio n w hich is re la te d to both th e p re v io u s th e o re m s . L et f be a b ije c tio n of the C“ R iem ann ian m anifo ld M w hich p r e s e rv e s the g e o d e s ic s of M. Is f n e c e s s a r i ly a d iffe o m o rp h ism ? To s t a r t w ith we m ake so m e re m a rk s w hich a r e in tended to th row ligh t on th e p ro b lem .

A r e a l p ro je c tiv e sp a ce ad m its a n a tu ra l C" s t r u c tu re and a R iem ann ian m e tr ic fo r w hich the l in e s a r e the m a x im a l g e o d e s ic s . F o r th is p a r t ic u la r c a s e , the fu n d am en ta l th e o re m of r e a l p ro je c tiv e g e o m e try im p lie s th a t th e a n sw e r is y e s , p ro v id ed th a t the d im en sio n of the sp a ce is > 1. On the o th e r hand , a s w as po in ted out to m e by T . P o s to n , it is v e ry e a sy to c o n s tru c t b ije c tio n s of th e un it s p h e re Sn c R n+1 w hich p r e s e rv e the g re a t c i r c le s and w hich a r e not h o m e o m o rp h ism s . F o r ex am p le , le t u s define x = (xi , . . . , x n+1) e Rn+1 to be ra t io n a l if a l l the n u m b e rs x¿ a r e ra tio n a l and to be i r r a t io n a l o th e rw ise . Then f: Sn -» Sn given by

f(x) = x, fo r x ra t io n a l , f(x) = -x fo r x i r r a t io n a l

is such a b ije c tio n .T he la s t exam ple show s th a t som e r e s t r ic t io n on f is n e c e s s a r y if it

is to be p roved to be a d iffe o m o rp h ism . And, of c o u rs e , the d im en sio n of M m u st be > 1. W ith th e se r e m a rk s a s a background we s ta te a p a r t ia l a n s w e r to the q u es tio n in the follow ing th e o re m .

T h e o re m . L et M be a C " R iem ann ian m anifo ld of d im e n sio n > 2 and le t f: M -* M be a h o m e o m o rp h ism w hich p r e s e r v e s the g e o d e s ic s of M.T hen f is a d iffe o m o rp h ism .

The r e s t r ic t io n d im M > 2 w hich a p p e a rs in th is th e o re m in p la ce ofd im M > 1 is i r r i ta t in g a s we do not th ink th a t it is n e c e s s a ry . H ow ever,it is p le a s in g th a t the th e o re m of M y e rs -S te e n ro d is a s im p le c o ro lla ry .F o r an is o m e try is a h o m e o m o rp h ism and it is not d ifficu lt to show th a t it p r e s e r v e s g e o d e s ic s .

In the p ro o f of the th e o re m we sh a ll m ake u se of sp e c ia l n e ig h b o u rhoods fo r the po in ts of M w hich we c a ll С-n e ig h b o u rh o o d s. To d e s c r ib e th e se ne ighbourhoods we c o n s id e r a po in t x e M and the ex ponen tia l m apping exp* of the tan g en t sp a ce Tx in to M. A n o rm a l neighbourhood of x is e x p xW w h ere

(i) exp is a d iffe o m o rp h ism on the open s e t W c T x ,(ii) if w e W then t w e W fo r a l l t , 0 s t S 1.

A С -neighbourhood U of x is a neighbourhood w hich is a n o rm a l n e ig h b o u rhood of each of i t s p o in ts . T hus any two po in ts of U can be jo ined by a un ique g eo d e sic se g m en t ly ing in U. It can be shown th a t e v e ry n e ig h bourhood of x co n ta in s a С-neighbourhood of x.

T he p ro o f p ro c e e d s in five s te p s w hich we sh a ll o u tlin e . F i r s t note th a t a g eo d e sic th rough a point x d e te rm in e s a unique d ire c tio n in th e p ro je c tiv e sp a ce Px a s so c ia te d w ith Tx . F o r although the g eo d es ic is the ra n g e of an in fin ite n u m b e r of g eo d es ic c u rv e s , th e tangen t v e c to rs to

IAEA-SM R-11/11 35

th e s e c u rv e s a t x a r e a l l n o n -z e ro s c a la r m u ltip le s of each o th e r . C onse q u en tly , a s th e h o m e o m o rp h ism f p r e s e r v e s g e o d e s ic s , it d e te rm in e s in an obvious w ay a function

fx: Px Pf(x)

fo r e a c h x e M . It is not d ifficu lt to show th a t f x is a h o m e o m o rp h ism .The second s te p is to e s ta b lish the follow ing te c h n ic a l le m m a . It is

an e x e r c is e in th e u se of the ex ponen tia l m ap.

L e m m a . L et U be a С -neighbourhood of x e M and le t a , b be g eo d e s ic s th ro u g h x in U. Suppose th a t a n , bn a r e se q u en c es on th e se g e o d e s ic s such th a t lim a n = lim b n = x. L et cn be a po in t on the un ique g eo d esic in U

jo in in g a n to b n , and deno te by yn the d ire c tio n a t x of the g eo d e sic in U jo in ing x to c n . T hen if lim yn e x is ts th is l im it l ie s on the lin e in Px

n-> »con tain ing the d ire c tio n s of a and b a t x. F u r th e r , e v e ry d ire c tio n on th is lin e can be ob ta ined a s such a lim it.

T he nex t s te p is to u se the le m m a to p ro v e th a t fx ta k e s lin e s in to lin e s . To do th is choose С -n eig h b o u rh o o d s U, V of x, f(x) re s p e c tiv e ly such th a t f(U )C V . T hen , g iven any two d ire c tio n s a, /3 in P x , le t a ,b be the g e o d e s ic s in U w ith th e se d ire c tio n s . L e t у denote a d ire c tio n on the lin e in P x con tain ing a, (3 and , in th e n o ta tio n of the le m m a , c o n s tru c t se q u e n c e s a n , b n , cn such th a t lim yn = y. The d ire c tio n fx(7n ) is the

n-» "d ire c tio n a t f(x) of the g eo d esic jo in ing f(x) to f ( cn) in V, and, fro m the co n tin u ity of fx , lim fx(Yn) e x is ts and is equa l to fx(y)- C onsequently ,

n-* "the le m m a m ay be ap p lied to the se q u e n c e s f(an), f(bn), f (c n) in V to show th a t f x(y) l ie s on the lin e in P f(X) con ta in ing f x(o') and fx ((3). It follow s fro m th e fu n d am en ta l th e o re m of r e a l p ro je c tiv e g e o m e try th a t, if d im M > 2 , then fx is induced by a n o n -s in g u la r l in e a r tr a n s fo rm a tio n be tw een Tx and Tf(Xj and is th e re fo re a d iffe o m o rp h ism .

T he fo u rth s te p is co n c ern ed w ith a lo c a l function w hich is like b ip o la r c o -o rd in a te s . L e t U be a С-neighbourhood and fix two po in ts y , z in U. G iven a po in t x e U (f y , z), th e re a r e unique g eo d e s ic s in U jo in ing x to у and x to z. T h ese g e o d e s ic s d e te rm in e d ire c tio n s in Py , PZ! re sp e c tiv e ly , and co n seq u en tly a function

Tyz : U - { y , z} - Py X P Z

can be defined in an obvious w ay. It is not d ifficu lt to show th a t Tyz is an im m e rs io n a t a l l p o in ts w hich a r e not on th e g eo d e sic in U con tain ing у and z.

T he la s t s te p is to show th a t f is d if fe re n tia b le a t a g e n e ra l point x e M . To do th is choose С -neig h b o u rh o o d s U, V of x, f(x) re s p e c tiv e ly su ch th a t f ( U) cV. F ix p o in ts y , z in U such th a t x is not on the g eo d esic in U con tain ing у and z. T hen on so m e neighbourhood of x,

rYZ ° f = (fy X fz)°Ty2

36 BR1CKELL

w h ere Y = f(y), Z = f(z). It fo llow s th a t th e co m p o sitio n tYz, " f is d if fe re n tia b le on th is neighbourhood and th e re fo r e , s in c e f is con tinuous and r yz ks an im m e rs io n , f a lso is d if fe re n tia b le . A s im ila r a rg u m e n t ap p lie s to f"1 and so f is a d iffe o m o rp h ism .

It i s p e rh a p s w orthw h ile po in ting out th a t the a rg u m e n ts we have u sed w ork in m o re g e n e ra l s itu a tio n s . F o r ex am p le , they apply to the g e o d e s ic s of any s p ra y on a d if fe re n tia b le m anifo ld .

IAEA-SM R-11 /12

STABILITY THEOREMS FOR R2-ACTIONS ON MANIFOLDS

C. CAMACHOInstituto de M atem ática Pura e A plicada,Rio de Janeiro,Brazil

Abstract

STABILITY THEOREMS FOR R2-ACTIONS ON MANIFOLDS.This paper introduces the notion of hyperbolicity for fixed points o f an R2-action on a manifold and

gives interpretation of this definition in terms of transversality. Hyperbolicity is a necessary condition of the stability of the germ of the action and it is also sufficient for lower dimensional manifolds. A fam ily of R2-actions generalizing the M orse-Smale flows is also defined. These actions exist on 2-m anifolds and Grassmann manifolds; moreover they are ^-stab le .

A n a c tio n of a L ie g roup G on a m an ifo ld M is a h o m o m o rp h ism tp: G ->• Diff(M) o f G in to th e g roup of d iffe o m o rp h ism s of M. T h is ac tion <p: g -*■ <pg i s ca lle d d if fe re n tia b le if th e induced m ap G X M -* M given by (g, x) -*• i/>g(x) is d if fe re n tia b le .

T h e o rb it th ro u g h a point x G M is th e se t o f p o in ts <pg(x) w h ere g G G and w ill b e deno ted by 0 x(tp). T he iso to p y g roup at x G M is defined a s Gx(<?) = i g G G; tpg(x) = x }. F ix in g a point x G M the m ap g -* <Pa(x) of G into M in d u ces an in je c tiv e im m e rs io n of the hom ogeneous sp ace d¡/Gx(tp) in to M w hose im age is 0 x(ip).

W hen G = R , the r e a l n u m b e rs , tp i s ca lle d a flow . By looking a t the c lo se d su b g ro u p s (Gx(tp) i s c lo sed ) o f R one concludes th a t any o rb it o f a flow is e i th e r a po in t o r h o m eo m o rp h ic to S1 o r a 1-1 r e g u la r im age of R.

A flow g e n e ra te s a v e c to r fie ld X on M by

X (x) = A ^ (x ) / t = о (*)

T hus tpt(x) can b e c o n s id e re d a s the so lu tio n of th e o rd in a ry d if fe re n tia l eq u a tio n (*) w ith in it ia l cond ition tp0(x) = x and th e o rb it th ro u g h x co in c id es w ith th e in te g ra l c u rv e of X p a s s in g th ro u g h x . C o n v e rse ly given a d if fe re n tia b le (say C1) v e c to r f ie ld X on a com pact m an ifo ld M, th e ex is te n ce and u n iq u en ess th e o re m fo r d if fe re n tia l eq u a tio n s and the c o m p a c tn e ss of M g u a ra n te e the e x is te n c e of a flow on M sa tis fy in g (*).

H e re we sh a ll be c o n c e rn ed w ith a c tio n s tp o f th e g roup G = R2, i . e . th e d ir e c t sum o f tw o c o p ie s of R . B y r e s t r ic t in g tp to each of th e s e cop ies one o b ta in s tw o flow s, say f and rj sa tis fy in g Çs rjt = r?tÇs fo r a l l s , t .

L e t X an d Y b e th e v e c to r f ie ld s g e n e ra te d by f and r). T hen the co m m u ta tiv ity cond ition on th e flow s m e a n s th a t the L ie b ra c k e t o f X and Y is id e n tic a lly z e ro ; one th e n sa y s th a t X and Y com m ute .

By looking at the c lo sed sub g ro u p s of R 2 one o b ta in s a l l p o ss ib le o rb its o f tp. If d im 0x(tp) = 0 th e n 0x(tp) is a po in t. If d im &x(cp) = 1 , 0 (tp) is h o m eo m o rp h ic to S1 o r R . If d im &x{tp) = 2, 0x(tp) i s hom eo m o rp h ic to S1 X S1, S1 X R o r R 2.

37

38 CAMACHO

E x am p le 1. T he R e a c t io n on th e p lane g e n e ra te d by two com m uting v e c to r fie ld s

X(x i< x 2) = (Xj+x2, Xg) and YfXj.Xg) = (Xj, xg)

h as th e fo llow ing o rb it s t r u c tu re : th e h a lf p la n es x 2 > 0 and Xg < 0 a re2 - o rb i t s , th e h a lf l in e s = 0, Xj < 0 and x 2 = 0, xx > 0 a r e 1 -o rb i ts , the o r ig in is a fixed po in t.

E x am p le 2. The R e a c t io n on 3 -s p a c e g iven by

XJXj, x2, x3) = ( -x 2,x i .x 3) and Y f x ^ x ^ x ) = ( x ^ x ^ O )

h as th e fo llow ing s t r u c tu r e . T he o r ig in i s a fixed po in t. T he h a lf lin e s x j = X2 = 0, х з> 0 and x : = x 2 = 0, x3 < 0 a r e 1 - o r b i t s . T he p lane XjX2 m inus the o r ig in is a c y l in d r ic a l o rb it . A ll o th e r o rb i ts a r e hom eo m o rp h ic to R 2 w ith s e t b o u n d ary {x3 = 0 } U { Xj = x2 = 0 }.

T he m o s t s im p le k in d of ac tio n s a r e th e l in e a r o n es , th a t i s , hom o- m o rp h is m s p : G ->■ A ut(E) of G in to th e g roup of l in e a r au to m o rp h ism s of a v e c to r sp a c e . B oth ex am p les above a r e l in e a r .

H y p erb o lic l in e a r a c tio n s . L et G b e iso m o rp h ic to Z o r R . T hen p is c a lle d h y p e rb o lic if fo r s f 0 a ll e ig en v a lu es of p(s) a r e d iffe ren t f ro m one in ab so lu te v a lu e .

Suppose now G = R 2 th en p i s ca lle d h y p e rb o lic if:(i) T h e re e x is ts a p - in v a r ia n t sp littin g of E , E = œ E t su c h th a t fo r any t ,

p i s t r a n s i t iv e on each connected com ponent of Et - 0.(N otice th a t (i) r e a d ily im p lie s th a t dim E t = 1 o r 2. ) L e t ir : E -> œ E t'

tVtb e th e p ro je c tio n m a p . F o r any n o n -z e ro v 6 E t th e m ap x t : Gy(p) -»Aut ( ® E .) defined a s Xf (g) ° 7r = 7r ° p(g) is by (i) th e ac tio n of a g roup

tV t 1iso m o rp h ic to Z o r R .(ii) F o r each t x t i s h y p e rb o lic .

E x am p le 3. E x am p le 1 is not h y p erb o lic ; how ever, a n e a rb y ac tio n g e n e ra te d by

X£( x i ,x 2) = (Xj+Xg+eXj, x 2) and Y(x1,x 2) = (x^ xg)

is h y p e rb o lic . M ore g e n e ra lly :T h e o re m : H y p erb o lic l in e a r R e a c t io n s fo rm an open and d en se su b se t

o f th e sp a ce of l in e a r R e a c t io n s .T h e re i s a lso an in te rp re ta t io n fo r h y p e rb o lic ity in te r m s of t r a n s

v e r s a l i ty given in the follow ingT h e o re m : L et E с A ut(E) be the s e t of n o n -h y p e rb o lic au to m o rp h ism s

of a v e c to r sp a c e E ( i . e . A E E if som e eigenva lue of A l ie s on the unit c i r c le ) . T hen p : R2 -» A ut(E ) i s h y p e rb o lic if and only if p in te r s e c ts E t r a n s v e r s a l ly out o f th e id e n tity .

A ssu m e f ro m now on th a t M is co m p ac t. L et xr (M) b e the sp ace of C r v e c to r f ie ld s w ith a Cr n o rm || . . . . || . A topology is in tro d u c e d in th e sp a ce o f a c tio n s a s fo llo w s. C a ll T^ : R 2 -» xr (M) the L ie a lg e b ra h o m o m o rp h ism in d u ced by <p. F ix a com pac t ne ighbourhood of z e ro К С R2 and define a C r d is ta n c e b e tw een two a c tio n s q> and Ф as

IA EA -SM R-11/12 39

An a c tio n (p i s ca lle d s t r u c tu ra l ly s ta b le if th e re is a ne ighbourhood N(<p) of (p in th e C? topology of a c tio n s such th a t fo r any Ф G N(<p) th e r e is a h o m eo m o rp h ism h : M -> M tak in g o rb its of ip onto o rb i ts of Ф. T he g e rm of y a t x i s ca lle d s tru c tu ra l ly s ta b le if fo r e v e ry neighbourhood U of x th e re i s a neighbourhood N(<p) of <p and fo r any € N(St') a po in t y £ U and a lo c a l h o m e o m o rp h ism h at x, h(x) = y, tak ing o r b i ts of ip onto o rb its of Ф.

D efin itio n . A fixed point x of an a c tio n <p: R 2 Diff(M) i s ca lle d h y p e r b o lic if th e l in e a r ac tio n on TXM defined a s g -* d<pg(x) i s h y p e rb o lic .

T h e o re m : L et x be a fixed po in t o f an a c tio n q>: R2 Diff(M ) su ch th a t th e g e rm of ip a t x is s t ru c tu ra l ly s ta b le . T hen x i s a h y p e rb o lic fixed point o f ip.

FIG. 1. Linear exam ples on Euclidean 3-space.

40 CAMACHO

T h e o re m : L et x be a h y p e rb o lic fixed point of an ac tio n Diff(M) su c h th a t d im M £ 4. T hen th e g e rm of gis a d iffeo m o rp h ism c o n tra c tin g a t x . T hen the g e rm of <p a t x is s t ru c tu ra l ly s ta b le .

T hus exam ple 1 i s not s t r u c tu ra l ly s ta b le . In fac t exam ple 3 is a n e a rb y a c tio n w ith d iffe ren t o rb it s t r u c tu re .

A ny s ta b le g e rm of R 2 ac tio n s on a 3 -m an ifo ld is eq u iv a len t to one of the l in e a r ex a m p le s in F ig . 1 on E u c lid ea n 3 -sp a c e . Out of the c o -o rd in a te p la n es a ll o rb i ts a r e h o m eo m o rp h ic to R2. T h e ir in te r s e c tio n w ith S2 is a lso show n.

T he no tion of h y p e rb o lic ity can be ex tended to com pact o rb its a s w ell. See R ef. [ 1] w here th is is done.

L e t {Kn} b e an in c re a s in g seq u en ce of com pac t n e ighbourhoods of ze ro c o v e rin g R 2. A point x e M i s ca lle d n o n -w a n d erin g if fo r any neighbourhoodV of x and n 0£ Z + th e re is g Í K n0 such th a t <pg(V) П V f 0 . We w rite £l((p) to deno te the se t of n o n -w an d erin g po in ts o f ip. An o rb it 0 x(<p) i s ca lled s in g u la r if Gx(<p) f 0. It is c le a r th a t any s in g u la r o rb it is con ta ined in £2(ip). In p a r t ic u la r any com pac t o rb it is in U((p).

T h e ac tio n ip i s c a lle d Í2 -s ta b le if th e re is a neighbourhood N(<p) such th a t fo r any Ф G N(<p) th e re e x is ts a h o m e o m o rp h ism h : П(<р) П(Ф) tak in g o rb its of <p onto o rb its of Ф.

In what fo llow s we c o n s id e r R2-a c tio n s s a tis fy in g the follow ing (i) T he n o n -w an d erin g se t is an em bedded c e ll com plex c o n s is tin g of

fin ite ly m any s in g u la r o rb i ts .(ii) T he com pact o rb its of tp a r e h y p e rb o lic .

( iii) T h e re is a o n e -p a ra m e te r su bg roup H C R 2 such th a t th e flow tp/H h asth e n o -c y c le p ro p e r ty .

(F o r th e defin ition of th e n o -c y c le p ro p e r ty se e R ef. [ 2] ) .T h e o re m : T h e re e x is ts on any 2 -m an ifo ld an R2-a c tio n tp s a tis fy in g

( i) - ( i i) - ( i i i ) .T h e o rb it s t r u c tu re g iven by th is th e o re m is a s show n in F ig . 2.T h e o re m : T h e re e x is t on the s p h e re Sn and the p ro je c tiv e sp a c e P"

R 2 -a c tio n s sa tis fy in g ( i ) - ( i i ) - ( i i i ) .

FIG. 2. Orbit structure given by theorem in text.

IA EA -SM R-11 /12 41

Indeed g iven a l in e a r a c tio n p on E n and the can o n ica l t r a n s i t iv e ac tio n ■iof A ut(E n) on th e G ra ssm a n n m an ifo ld Gm, n -m one o b ta in s R e a c t io n s p on G m ,n -m by th e follow ing d ia g ra m :

R\

\

\

A ut(E n)

■ф

Diff(Gm , n -m )

One p ro v e s th a t u n d e r c e r ta in cond itions on th e e ig en v a lu es of p, p s a t is f ie s( i) - ( ii) - ( ii i ) -

A s an i l lu s tr a t io n we d e s c r ib e an R -a c tio n on S3 g iven by th is th e o re m :L e t p : R2 -» Aut(R4) b e defined by p ís j , s2) = e x p ^ X ^ s 2X 2) w here

X j = diag(X, X, X3, X4) X4 < X < X3 and X2 = diag(,u, M2. M3, M ) M3 < U < M2.T h e o rb it s t r u c tu re is as fo llow s. The in te r s e c tio n s of the x ¡ -a x is w ith

S3 y ie ld eigh t h y p e rb o lic fixed po in ts of "p. T he in te r s e c tio n s of th e x¡Xj-p la n es of R4 w ith S3 g ive the s e t of n o n -w a n d e rin g po in ts of ~p. So Q(p) =

6U Г2 w h ere each S3, is a 1 - s p h e re ,

t = 1 t t ^S uppose Í21; Í32, £33 a r e the 1 - s p h e re s p a s s in g th ro u g h th e fixed p o in ts

of "p ly ing on the x j - a x is . T he in te r s e c t io n s of th e 3 -p la n e s XjXjX of R 4 w ith S 3y ie ld fo u r p - in v a r ia n t 2 - s p h e r e s . Out of th e s e s p h e re s a l l o rb its of ~p a r e h o m eo m o rp h ic to R 2 w ith bou n d ary Í34 U Q5 U Щ .

T h is exam ple as w ell a s th e ex a m p le s on 2 -m an ifo ld s a r e s tru c tu ra l ly s ta b le . In fac t one h a s th e follow ing th e o re m s :

T h e o re m . R e a c t io n s on a 2 -m an ifo ld sa tis fy in g (i)-( ii) a r e s t ru c tu ra l ly s ta b le [ 3 ] . оT h e o re m . R -a c tio n s on 3 -m an ifo ld s induced by two com m uting g rad ie n t v e c to r f ie ld s and sa tis fy in g ( i)-( ii) a r e s t r u c tu ra l ly s ta b le .

T h e follow ing £3-s ta b il i ty th e o re m fo r R e a c t io n s i s a g e n e ra liz a tio n of a th e o re m of J . P a l is fo r d iffe o m o rp h ism s (see R ef. [4] ).

T h e o re m ([ 3] ). R e a c t io n s on an n -m a n ifo ld s a tis fy in g ( i) - ( ii) - ( ii i ) a r e Í3 -s ta b le .

R E F E R E N C E S

[1] C A M A C H O , C . , On Rk X Z'^-actions, Proc. Symp. Dynamical Systems, Salvador, Brazil (1971).[2] SMALE, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747-817.[3] C A M A C H O , C . , Morse-Smale Reactions on 2-manifolds, Proc. Symp. Dynamical System, Salvador,

Brazil (1971).[4] PALIS, J., On Morse-Smale dynamical systems, Topology, _8 (1969) 385-405.[5] C A M A C H O , C . , An instability theorem for Reactions to be published.

IA EA -SM R-11/13

INTRODUCTION TO MINIMAL-SURFACE THEORY

E. DE GIORGIScuola N orm ale Superiore, Pisa,Italy

Abstract

INTRODUCTION TO MINIMAL-SURFACE THEORY. Definitions of Hausdorff measure and of perim eter are given, followed by applications to a generalized problem o f m inim al surfaces.

T he a im of th is co n trib u tio n is to p ro v id e in an in fo rm a l way som e co n cep ts about m in im a l- s u r fa c e th e o ry . We f i r s t c o n s id e r the follow ing " D ir ic h le t p ro b lem ":

L et B j be the un it s p h e re of IRn, i. e . B j = { x E ]Rn: | x | < 1} . L e t f be a r e a l-v a lu e d function defined on the bou n d ary of В ^Э В ^; we suppose th a t f h as a c e r ta in d e g re e of r e g u la r i ty (for ex a m p le , C ^B B j)). The p ro b lem is th u s th e follow ing: find a function g G C1(B1) su ch th a t g G C°(B1)J g = f onSBj and

i/ ( l + | V g | V dx g ^ ( l + | V h | 2) dx (1)Bi Bi

fo r any h G C1(B1) П C°(B1) su ch th a t h = f on 3Bj.It h a s been p ro v ed th a t th is p ro b le m has a unique so lu tio n , and th a t th is

so lu tio n is a n a ly tic in Bj. One could a ls o w eaken the cond itions on f, ju s t r e q u ir in g th a t f be con tinuous. F u r th e rm o re , the th e o re m does not only ho ld tru e fo r a s p h e re , bu t a lso fo r any open dom ain (with sm o o th boundary ) w ith the bou n d ary hav ing p o sitiv e m ean c u rv a tu re . F o r th e se r e s u l t s , s e e a v e ry c le a r ex p o s itio n in Ref. [3].

We m u st now re m a rk th a t if we re m o v e the cond ition th a t the b oundary of the dom ain h as p o s itiv e m ean c u rv a tu re , we can no lo n g e r expect to find a so lu tio n w hich is con tinuous up to the bou n d ary . T he follow ing s im p le p ro b le m show s w hat happens:

L et D = {x G ]Rn: 1 < | x |< 2} . L e t f be 0 fo r | x | = 2 , and M fo r | x | = 1. T hen , fo r la rg e M, th e re e x is ts no so lu tio n of the D ir ic h le t p ro b lem fo r m in im a l s u r fa c e s in the se n se d e s c r ib e d above.

It is th e re fo re n e c e s s a ry to g e n e ra liz e the concep t of s u r fa c e , and , of c o u r s e , the concep t of a r e a in o r d e r to be ab le to so lv e m o re g e n e ra l p ro b le m s .

F i r s t of a l l , we need a su ita b le (n-1 ( -d im e n s io n a l m e a s u re . T h is w ill be the H ausdo rff m e a s u re : fo r any in te g e r k , O S k S n , fo r any s e t E £L]Rn, we define

CO

Hk(E) = sup 12 u k in f \ (diam E h)k; U E . D E , d i a m E . < e e>0 L h=l

(2)

h = 1

w h ere cok is the k -d im e n s io n a l L eb esg u e m e a s u re of the un it s p h e re in Шк.

43

44 DE GIORGI

It is e a sy to p ro v e th a t Hk is a B o re l r e g u la r m e a s u re , i. e. a l l B o re l s e ts of IRn a r e m e a s u ra b le 1 and fo r any se t E th e re e x is ts a B o re l se t В Э E su ch th a t Hk(B) = Нк(Е). F u r th e rm o re , on r e g u la r m an ifo ld s (C1-m a n ifo ld s , fo r exam ple ) Hk is the sa m e as the com m on k -d im e n s io n a l m e a s u re .

Now le t A be an open se t of IRn; fo r any function g : A s^ -R , we define the s e t E(g) by the follow ing re la tio n :

E(g) = | ( x , z) G IR X R : x G A, z S g (x)j- (3)

We com e back to the c a s e in w hich f = 0 fo r |x | = 2, f = M fo r | x | = 1. We choose the open s e t A such th a t A D D = {x G IR11: 1 S | x | S 2} , and we t r y to m in im iz e

Hn(8E(g) П A)

when

g = 0 fo r x G А П - |x G ]Rn: | x | ë 2

g = M fo r x G A Pi -j x G IRn: l x l S i

It is c le a r ly a re a so n a b le g e n e ra liz a tio n , but the H ausdo rff m e a s u re is not too su ita b le if we w ant to u se the d ir e c t m ethod of the c a lc u lu s of v a r ia t io n s . We sh a ll th e re fo re g ive a d if fe re n t defin itio n of " p e r im e te r " of a (B ore l) se t of IRn.

L et A be an open se t of IR11. F o r any B o re l s e t В £L IR11 we define thep e r im e te r of В w ith r e s p e c t to A by the follow ing re la tio n

P (B , A) = sup J d iv g dx; g G [c J(A )]n , | g | Ê lj-

AnB

If P (B , A) < + oo we sa y th a t В has fin ite p e r im e te r w ith r e s p e c t to A. If P (B ,K ) < + со fo r any open se t К su ch th a t К is a co m p ac t s e t con ta ined in A(KCCA) we say th a t В h as lo c a lly fin ite p e r im e te r w ith r e s p e c t to A.

Now fo r any B o re l s e t В le t B 1/2 denote the s e t2

x G IR0 : lim [u p11]’1 m e a s [A (x) П B] = 1 /2 p—* 0 n P

if the s e t В h as lo c a lly fin ite p e r im e te r w ith r e s p e c t to A, then

Р ( В ЛА) = Нп_х(В1/2 П A)

1 A set В is said to be measurable with respect to an exterior measure о if for any subset E of the space it holds that о(Е) = о(ЕПВ) + o(E-B).

2 Ap(x) ={y e R n : I y - x I < p } .

IAEA-SM R-11 /13 45

In p a r t i c u la r , if В has a C1-b o u n d a ry , then

P (B , A) = (ЭВ П A)

The fo llow ing two th e o re m s allow u s to u se the d ir e c t m ethod of the c a lc u lu s of v a r ia t io n s in p ro b le m s w h ere the p e r im e te r is involved:

1. L et A be an open se t of IRn. {Bj,} is a seq u en ce of B o re l s e ts . If fo r any s e t KCC A th e re e x is ts a c o n s ta n t 7 (K) su ch th a t

P ÍB j^K ) á y (K) fo r any h

th en th e re e x is t a su b seq u en ce Bh and a B o re l s e t В su ch th a tк

' Bh ^ B 111 L loc(A ) к

( i .e . fo r any К CCA

lim m e a s k-»°°

Bu - В U В - B,uk / J

П К 0 )

then2. If a seq u en ce of B o re l s e ts {B^} ten d s to a B o re l s e t В in L* (A),

P (B , A) § lim inf P (B , A) h->»

T h e re a r e m any d if fe re n t w ays of g e n e ra liz in g the m in im a l-s u r fa c e p ro b le m . T he u se of p e r im e te r is t r e a te d in Ref. [2]; fo r o th e r m ethods c o m p a re Ref. [1 ] . F u r th e r r e f e r e n c e s can be found in R efs [1 -3 ] .

R E F E R E N C E S

[1] ALMGREN. F .J . J r .. ta lk a t present Summer C ollege.[2] DE GIORGI. E ., COLOMBINI, F ., PICCININI, L .C ., Frontière orientate di misura m inim a e problem i

connessi, Edizioni Scuola Normale Superiore (Pisa 1972).[3] MIRANDA. М .. Paper to appear in the "Lecture Notes of the CIME Session on Geometric.M easure Theory

and M inimal Surfaces".

IA EA -SM R-11 /1 4

ON COMPLEX VARIETIES OF NILPOTENT LIE ALGEBRAS (AFTER G. FAVRE)

P .DE LA HARPE M athem atics Institute,University o f Warwick,Coventry, Warks,United Kingdom

Abstract

ON COMPLEX VARIETIES OF NILPOTENT LIE ALGEBRAS (AFTER G. FAVRE).It is well-known that there are uncountably many isomorphism classes o f nilpotent Lie algebras; hence

one expects classification results ( if any) to be given in terms of continuous param eters. This paper gives exam ples proving that such (partial) results can indeed be obtained in a few cases. Sections 1 to 4 illustrate, through nilpotent Lie algebras of m axim al rank, the methods of Favre. This is a preparation for the two exam ples o f section 5, where fam ilies of nilpotent Lie algebras are param etrized by varieties; these are quotients of projective spaces by finite (possibly trivial) groups of transformation.

1. MOTIVATIONS

A ll L ie a lg e b ra s c o n s id e re d h e re a r e defined o v e r the com plex f ie ld C, and th e y a r e of f in ite d im e n sio n w ith the only ex cep tio n of th e f re e a lg e b ra Lie(w) and of i ts id e a ls a p p e a r in g in exam ple 6. The s e t o f s t r ic t ly p o s itiv e in te g e rs is deno ted by N* = { 1 , 2 , . . . } .

L et _g be an a r b i t r a r y L ie a lg e b ra . I t is a c la s s ic a l r e s u l t th a t £ can be w ritte n a s a s e m i- d i r e c t p ro d u c t j j = rx ,,¿ w h ere ф is an h o m o m o rp h ism fro m _s to the L ie a lg e b ra D e r(r) o f d e r iv a tio n s of jr, w h ere the d e r iv e d id e a l o f £ is n ilp o ten t and w h ere s is a d ir e c t p ro d u c t of s im p le L ie a lg e b ra s :

к_s = П s¡ (see B ourbak i [1(1, 5 and 6)] and [2]).

i = l

M uch is known about s e m i- s im p le L ie a lg e b ra s su ch a s s¡; in p a r t ic u la r , th e i r fu ll c la s s if ic a t io n h as b een ach iev ed by E . C a rta n (a fa s t in tro d u c tio n to th is th e o ry is p ro v id ed by S am elso n [3]). We s h a ll r e ta in th e two fo llow ing fa c ts :

(1) T h e re a r e coun tab ly m any iso m o rp h is m c la s s e s only of s e m i-s im p leL ie a lg e b ra s . k

(2) Any s e m i- s im p le L ie a lg e b ra _s is a d ir e c t p ro d u c t _s= П s¡ as

above and is co m p le te ly c h a r a c te r iz e d (up to iso m o rp h ism ) by a fin ite s e t of d is c re te p a r a m e te r s (fo r ex am p le : the n u m b e r к in the p ro d u c t above, and the ra n k and type of each of th e Si ).

In c o m p ariso n , v e ry l i t t le is known about the s t r u c tu re of n ilp o ten t L ie a lg e b ra s ; in p a r t ic u la r , a fu ll c la s s if ic a t io n s e e m s to be out o f r e a c h at p re s e n t. H ow ever, so m e r e s u l t s have been r e c e n tly ob ta ined by v a r io u s a u th o rs inc lud ing A m iguet [4] and F a v r e [5]. A p a r t ia l r e p o r t on th e ir

47

48 DE LA HARPE

w ork lim ite d to the s tudy of n ilp o ten t L ie a lg e b ra s _g of m a x im a l ra n k , as defined in se c tio n 3, is given in th is p a p e r . (R e fe re n c e s [4, 5] ob ta in , ho w ev er, m o re ex ten s iv e in fo rm a tio n bo th on £ and on D er(g), and w ithout the r e s t r ic t io n of m a x im a l ran k . )

2. SYSTEMS OF WEIGHTS

L e t £ be an a r b i t r a r y L ie a lg e b ra and le t D er(g) be the L ie a lg e b ra of i ts d e r iv a tio n s . A m a x im a l to ru s fo r £ i s by defin ition a su b a lg e b ra Tjg of D er(g) w hich is ab e lian , w hich c o n s is ts of s e m i- s im p le d e r iv a tio n s (i. e. the l in e a r m ap Д :£-* £ can be d iag o n alized fo r each Д е Т ^ ) , and w hich is m a x im a l am ong su b a lg e b ra s of D er(g) hav ing th e se two p ro p e r tie s . L e t Tj[ be a fixed m a x im a l to ru s fo r £ and le t T*jj be its dual (as v e c to r sp a ce ). C o n s id e r the n a tu ra l r e p re s e n ta t io n of Tg in _g, w hich is induced by the ev a lu a tio n m ap D er(g) Xj*-> _g. F o r each e igenva lue a e T * g , le t S°' be th e e ig e n sp a c e

jx e _ g /A (X ) = a(A)X fo r a ll A e T g j

L e t R_g(Tj[) be the (finite) s u b se t o f th o se <* fo r w hich d im (g f )> 0. As the e le m e n ts of T g fo rm a com m uting fam ily of d ia g o n a lizab le en d o m o rp h ism s of th e v e c to r sp a ce £, th e a lg e b ra £ can be w ritte n as

£ = 0 £ a aeR g(T g)

By defin itio n , the sy s te m of ro o ts defined by T g is the s e t Rj?(Tj?)={a'eT*g/dim (gtI)> 0}; the s y s te m of w eigh ts defined by T g is the s e t Pg(Tg) = {(a, d a )e R £ (T £ )X N * /d « = d im (g“ )}.

It happens th a t th e se sy s te m s do not depend e s s e n tia l ly on the choice of Tg. M ore p re c is e ly , le t £ and ¿ be two L ie a lg e b ra s w ith m a x im a l to r i Tj[ and T 'g 1 . and le t Rj[(Tj[) and R g 1 (T 'g 1) ( re sp . Pj*(Tj;) and P g '(T 'g ') ib e the c o rre sp o n d in g s y s te m s of ro o ts ( re sp . w eigh ts); then P_g(Tg) and P j f '( T '¿ ) a r e sa id to be eq u iv a len t if th e re e x is t l in e a r b isec tio n s cp : Tg -> T 'g ' and ф :£ such th a t the d ia g ra m

T g X £ -------------------------------------£

Ф X ф j фT

T '¿ _ X £ l -----------------------

co m m u tes (o th e rw ise sa id : if th e re p re s e n ta t io n s of Tg in £ and of T 'g ' in £l a r e eq u iv a len t). But now, in c a se g = g' :

P ro p o s itio n 1. L e t T g and T'_g be two m a x im a l to r i fo r _g. T henth e re e x is ts an au to m o rp h ism в of £ su ch th a t 0(Tg) = T 'jg, w h ere V is ther e s t r ic t io n to Tjj of the a u to m o rp h ism

Der(_g)---------------► Der(_g)

A '--------------- -в-А-в ' 1

IA EA-SM R-11 /14 49

P ro o f: It can be re a d out of a r e s u l t due to M ostow [6 , ( th eo rem 4.1)], and can a lso be found in A m ig u e t's th e s is [4 (4, th e o re m 16)] o r F a v r e 's th e s is [7 (1, th e o re m 2)].

It follow s fro m p ro p o s itio n 1 th a t, keep ing the sam e n o ta tio n s , the d ia g ra m

T gX .g ---------------------------------- » £

0X 0

T > _ g X £ ------------------------------------------------------► £

co m m u tes , so th a t Pjg(Tj[) and P.g(T'_g) a r e equ iva len t. T hey can be_ f P g (T 'jg )--------► P£(T£)

id e n tif ied by 0 - , w here '■'в: T'*_g-> T*_g is the( o ' , d o ' ) i-------*' ( t0 ( a l), d a 1)

tr a n s p o s e d m ap of 0. H ence, we w ill w rite P g ( re sp . Rg) in s te a d of Pg(Tg) ( re sp . Rg(Tj*)) and c a ll i t the sy s te m of w eigh ts ( re sp . of ro o ts ) o f the L ie algebra_g .

E xam ple 1. L e t _g be a s e m i-s im p le L ie a lg e b ra . T hen D er(g) is can o n ica lly iso m o rp h ic to £ so th a t:

(a) m a x im a l to r i fo r _g can be id en tified w ith C a rta n su b a lg e b ra s of £;(b) the ro o t sy s te m of _g defined above co in c id es w ith the ro o t sy s te m

u su a lly defined.

N ote th a t in th is c a se d a = 1 fo r each n o n -z e ro ro o t a and d a is equal to the ra n k of _g when a = 0.

E x am p le 2. L e t £ be an ab e lian L ie a lg e b ra . T hen D er(g) is the L ie a lg e b ra _gl(_g) of a l l l in e a r m aps fro m _g into i ts e lf . L e t (e¡)i e I be a b a s is

fe ¡ if j = кof £ and le t (E¡ j)¡ j be the u su a l m ap s defined by E ; . (ek ) = -

o th e rw ise .Then T£= ф C E U is a m a x im a l to ru s fo r £. F o r each j e I, le t

i e I ' [ T g = © C E U ---------- « - C

Qfj-e T*g be the m ap 4 ’ . Then Rg = e T - ^ g / je l} and

daj = 1 fo r ea ch j e l .EXi E u '-------

E x am p le 3. L e t _g be a n ilpo ten t L ie a lg e b ra such th a t ev e ry d e riv a tio n of £ is n ilp o ten t. (Such a lg e b ra s do e x is t: se e B ourbak i [1 (4, ex. 19 and5, ex. 13)] o r F a v re [8]). Then th e only m a x im a l to ru s fo r £ is {0}, so th a t P_g is a s e t w ith one e lem e n t: P_g= {(0, dim(g))}.

E xam ple 4 . L et £ be a s e m i- s im p le L ie a lg e b ra , le t jd be a p a ra b o lic su b a lg e b ra of £ and le t _b be a B o re l su b a lg e b ra of £ con ta ined in £ . Both £ and b have in n e r d e r iv a tio n s only and have z e ro c e n tr e s (see F a v re [7 (4. 4)] and [9]). Knowing th a t, one can e a s ily d e s c r ib e the sy s te m s of w eigh ts of £ and of b in te rm of th e ro o ts of £; d e ta ils a r e le ft to the r e a d e r (see a lso ex am p le 5 below ).

50 DE LA HARPE

3. N ILPO TEN T LIE ALGEBRAS

Let_g be a L ie a lg e b ra and le t (Cn_g)neN=;t be i ts lo w er c e n tra l s e r ie s , defined by Cx_g=¿ and C n+1_g= [_g, Cnj?] when n e № !. T hen _g is n ilp o ten t if Cn_g= {0} fo r n la rg e enough. If _g is a n ilp o ten t L ie a lg e b ra , th e re a re th r e e im p o rta n t in v a r ia n ts a ttac h ed to _g:

T he type s of g is the cod im en sio n of C 2g inThe n ilp o ten t c la s s (or s im p ly c l a s s , below) is the la r g e s t in te g e r p

such th a t C P j;/ {0}.T he ra n k к of ¿ is the d im en sio n of the m a x im a l to r i fo r _g.

P ro p o s itio n 2 . L e t | b e a n ilp o ten t L ie a lg e b ra , and le t s , p, к be as above. Then:

(a) any co m p lem en t w of C2g in ¿ g e n e ra te s £ as a L ie a lg eb ra ;(b) any s e t of g e n e ra to rs fo r ¿ co n ta in s a t le a s t s e lem e n ts ;(c) if w is any com plem en t of C2¿ in two d e r iv a tio n s of _g w hich

co incide on w a re id e n tic a l;(d) the in eq u a lity OS k s s ho lds.

P ro o f: E ac h p a r t o f p ro p o s itio n 2 is a re a so n a b ly e a sy e x e rc is e , le f t to the r e a d e r ; a l te rn a tiv e ly , se e A m iguet [4 (3, th e o re m 9, and4, le m m a 9)].

T he g e n e ra l p ro g ra m m e now is to apply the no tions in tro d u ced in se c tio n 1 to n ilp o ten t L ie a lg e b ra s . In o r d e r to m ake life e a s ie r , we r e s t r i c t o u rs e lv e s fro m now on to n ilp o ten t L ie a lg e b ra s of m a x im a l ran k , n am ely to th o se fo r w hich k = s (no tations as in p ro p o s itio n 2d). I t is show n in R ef. [5] th a t p a r t (but no t all!!) of the r e s u l t s below c a r r y o v e r to a r b i t r a r y n ilp o ten t L ie a lg e b ra s . N ote th a t n ilp o ten t L ie a lg e b ra s of m a x im a l ra n k and L ie a lg e b ra s behav ing as th a t o f exam ple 3 a re the two e x tre m e c a s e s : к = s and к = 0.

We w ill now s ta te som e p ro p e r t ie s of a lg e b ra s of m a x im a l ra n k , and then " p ro v e " th e m by v e r if ic a tio n on the ex a m p le s o f the nex t sec tio n . H onest p ro o fs a r e given in Ref. [5 (I)] .

D efin itio n . L e t R be a f in ite s u b se t of a v e c to r sp a ce V w hich sp a n s V, and le t В be a su b se t of R (think of Rg in T*g, w ith a L ie a lg eb ra ) .

(a) В is sa id to be a b a s is o f R if it is a b a s is o f V in the u su a l se n se and if m o re o v e r any e lem e n t of R can be w ritte n as a l in e a r com bination of e le m e n ts in В w ith p o s itiv e in te g e r co e ffic ien ts .

(b) L e t В be a b a s is of R; a В-p a th in R is a sequence (0i, (32 , . . . , (3q ) of e le m e n ts of R su ch th a t, fo r each i e { l , 2, . . . , q - 1}, e i th e r -|3 i+1e B o r /3i+1 - / 3 i e B .

(c) L e t В be a b a s is o f R; a B -co n n ec ted com ponent o f R is an a rc -c o n n e c te d com ponent of R fo r the p re v io u s defin itio n of B -p a th s .

P ro p o s itio n 3. L e t g be a n ilp o ten t L ie a lg e b ra of type s, o f c la s s p and of m a x im a l ra n k ; le t Tj* be a m a x im a l to ru s fo r _g; then the sy s te m of w eigh ts P_g en joys the follow ing p r o p e r t ie s :

IAEA-SM R-11 /1 4 51

(a) Rg co n ta in s a un ique b a s is В = { a ,, . . . ,a } .s s

(b) L e t a = ^ п ;а ; e Rg, and le t |e | = ^ rijbe i ts leng th ; theni= 1 i = 1

l a |a I S p. F o r each i e {1, . . . , s} , the only m u ltip le of w hich is in R g is Oj i ts e lf .

(c) T h e re is a function f of s in te g e r v a r ia b le s , ta k in g in te g e r v a lu es ,s

and su ch th a t the follow ing ho ld s: if a = ^ is a s in (b), theni = l

d a = d im fg” ) 5 ffnj, . . . , n s).(d) L e t (Rj_g)j = j { be the B -co n n ec ted com ponen ts of Rg and le t

gj = © . g“ fo r each j e { 1, . . , I }.aeRJ£

T hen: each is a n ilp o ten t L ie a lg e b ra of m a x im a l ra n k , say of type Sj and of c la s s Pj ; the e q u a litie s

ii

Sj and p = sup (pj )

i = i j = 1

hold; m o re o v e r gj cannot be w ritte n as a d ire c t p ro d u c t o f two L ie a lg e b ra s , th is fo r each j e l l , . . , SL}.

N. B. : th e b e s t p o ss ib le function f fo r (c) is given by the d im en sio n s of the ro o t sp a c e s of th e ad hoc m odel (exam ple 6), and can be e x p re s se d in te rm s of the M obius function . (It is obv iously th e s a m e function as th a t in B ourbak i [10 (II. 3, fo rm u la 16)).

4. EX AM PLES OF W EIGHT SYSTEMS

E x am p le 5. L e t g be a s e m i- s im p le L ie a lg e b ra , le t h_be a C a rtan su b a lg e b ra of g , le t á? be the s e t of n o n -z e ro ro o ts of g w ith r e s p e c t to h, le t & = 3t*U ¿ft " be a p a r t i t io n o fá ? in p o s itiv e and n eg a tiv e ro o ts , and le t

r rT = 0 g ?g = n l 0 h ® n w ith -i ae.31

I n = 0 £ ?C(Ë +

be a C a r ta n d eco m p o sitio n of g . Then n is c le a r ly a n ilp o ten t L ie a lg e b ra w hose type is the ra n k of _g.

Any e lem e n t X of h d e fin es a d e r iv a tio n

is an in je c tio n h -» D er(n); and it is c le a r th a t the im a g e of h by th is in je c tio n is a m a x im a l to ru s fo r n. H ence n is of m a x im a l ran k . The p ro o f of p ro p o s itio n 3 fo r th is exam ple fo llow s s tra ig h tfo rw a rd ly fro m the th e o ry of s e m i- s im p le L ie a lg e b ra s ; in p a r t ic u la r , Rn=¿¿?+ and d o = d im (g a ) = 1 fo r ea ch a e Rn. (F ig . 1).

j y i ___► [x Y ]’ So th a t th e re

52 DE LA HARPE

V a 1+a 2

FIG. 1. Rn for g= s 1(3,0).

A 3 -d im e n s io n a l h a rd w a re m odel fo r Rn when _g= so (7 , C) looks n ice , though u s e le s s fo r so a p -b u b b le s .

E x am p le 6. L e t u s t r y and c o n s tru c t the n ilp o ten t L ie a lg e b ra of type s and c la s s p, w hich has "a s few e x t r a r e la t io n s a s p o s s ib le " . L e t w be a v e c to r sp a ce of d im en sio n s, le t ® w be its te n s o r a lg e b ra , and le t L ie (w) be the L ie a lg e b ra g e n e ra te d by w in ® w . Though L ie (w) is in fin ite d im e n s io n a l as soon a s s ë 2, the quo tien t m (s , p) = L¿e(w)/CI>+1(Lie(w)) is f in ite d im e n sio n a l fo r each p e № ; it is by defin ition the m odel of type s and of c la s s p. (E x e rc ise : show th a t in (s ,p ) h as indeed type s and c la s s p .) The in te g e rs s and p being fixed once and fo r a ll, le t u s w rite in in s te a d of m (s , p).

We iden tify w w ith a co m p lem en t of C2 in in m and we choose a b a s is (e¡)i=i _ # s of w. F o r each i e { l , . . . , s } , le t Д ¿ be th e d e r iv a tio n of in w hich ex ten d s the l in e a r m ap

w --------------------------*■ in

e . ,__________________ ► \ e i if J = iJ [O o th e rw ise

SThen T m = ф С Д , is a m a x im a l to ru s fo r m , and in p a r t ic u la r m. is of

i— 1m a x im a l ran k .

L et (аь . . . , a s) be th e b a s is o f T *m w hich is dual to th e b a s is (Aj , . . , A s) of Tm . Then

s n ¡e N fo r each i = 1, . . , sS

a = ^ njQ-j e T * m 1 § ^ n¡ = O' 1 È pi “ 1 i = 1

the only m u ltip le of a¡ w hichis in R m is i ts e lf , fo reach i = 1, . . , s.

as one can show by w ritin g a b a s is of rn in te rm of the e ¡ 's and of th e ir b ra c k e ts .

D ig re ss io n . The c o n s tru c tio n of exam ple 6 is of a s ty le w hich is fa ir ly s ta n d a rd in m any do m ain s. In the p a r t ic u la r ca se of n ilp o ten t L ie a lg e b ra s of given type and c la s s , it s e e m s to go back a t le a s t to S c o rz a [11]. The d e r iv a tio n s of th e m odel have been s tu d ied by S chenkm an [12]. The

IAEA -SM R -11/14 53

m odel h as been u se d by Sato [13], and by D y ers fo r c o n s tru c tin g h is exam ple given in [14]. A cco rd in g to the m o tiv a tio n s w hich in tro d u c e ex am p le 6, any n ilp o ten t L ie a lg e b ra of type s and c la s s p is iso m o rp h ic to a quo tien t m (s , p ) / a w here a is an id e a l in the m odel such th a t CP rn(s, p) ^ a C C 2m( s , p). A m iguet p ro v e s th e in te re s t in g follow ing r e s u l t (see [4 (3, th . 12)]).L e t m .= m.(s ’ p) be as above, le t a. and a¿ be two id e a ls of in such th a t

a n d ¿ = m / a ¿ a re both of type s and of c la s s p; then g and g' a re iso m o rp h ic if and only if a. and a^ a re conjugate u n d er an a u to m o rp h ism of in. One consequence of th is is a se ttin g fo r c la s s if ic a t io n p ro b le m s (at given type and c la s s ) :

To a fam ily of ( iso m o rp h ism c la s s e s of) L ie a lg e b ra s c o rre sp o n d s a fam ily of id e a ls in m , n am ely a su b se t of a G ra s sm a n n m an ifo ld . The g roup Aut(m) a c ts on th e s e id e a ls , and th e re is a o n e - to -o n e c o rre sp o n d e n c e b etw een o rb its fo r th is ac tio n and L ie a lg e b ra s in the given fam ily . C o n s id e ra tio n s w ith w eight sy s te m s m ake it then u se fu l (and often even su ffic ien t) to s tudy the o rb its defined by a conven ien t ac tio n of a fin ite p e rm u ta tio n group on som e of th e se id e a ls , in s te a d of study ing the ac tion of the " la rg e " group Aut(m) on a ll o f them . The ex a m p le s of se c tio n 5 w ill hopefu lly i l lu s t r a te the m ethod b e t te r than any fu r th e r com m ent. N ote, how ever, th a t the " re d u c tio n " of the p ro b le m w aved a t above, w hich allow s one to c o n s id e r g ro u p s of p e rm u ta tio n s in s te a d of g ro u p s of au to m o rp h ism s of m o d e ls , depends s tro n g ly on the fac t th a t one looks h e re at a lg e b ra s of m a x im a l rank .

E x am p le 7 . L e t rn = rn(3, 3) be the m odel o f type 3 (F ig . 2) and of c la s s 3, and le t e 1( e 2, e 3 be a s in exam ple 6. T hen in has the follow ing b a s is :

e j , e 2, e 3 w hich span w;

e 7 = [[e2 , e j , e3 ] e g = [[e3 , e j , e2 ]

e 9 = [[e2 , e j , e j e 10 = [[e2 , e j , e 2]-w h ich span C3 m-w h ich span C3 m

en = t [ e 3, e j , e j e 12 =[ [ e 3, e j , e 3]

e i3 = [[е з > e 2 e 2 e 14 = [ [e 3, e 2] ,e 3]

The ro o t s y s te m fo r in is given by Rm = B U R®m U R® in U R® m w h ere

В ={«!■ а2, o 3 >

54 DE LA HARPE

j>a2+2a3i

5. VARIETIES OF N IL PO TE N T LIE ALGEBRAS

T he p ro b le m w hich th is se c tio n i l lu s t r a te s is the follow ing: L e t g be a n ilp o ten t L ie a lg e b ra of m a x im a l ra n k , and le t P g be its sy s te m of w eights. C la ss ify a ll n ilp o ten t L ie a lg e b ra s of m a x im a l ra n k having the sa m e sy s te m of w eigh ts.

As a f i r s t ( tr iv ia l) ex am p le , le t g be of m a x im a l ra n k and suppose th a t P g is eq u iv a len t to P m (s . p); then _g is iso m o rp h ic to th e m odel in (s , p). S ection 2. 7 of R ef. [5] p o in ts out a r ic h c la s s o f o th e r L ie a lg e b ra s of m a x im a l ra n k w hich a r e co m p le te ly c h a ra c te r iz e d (up to iso m o rp h ism ) by th e ir sy s te m s of w eigh ts. A cco rd in g to the t i t le o f the p re s e n t se c tio n , we give now two ex am p les of a d iffe re n t kind.

E xam ple 8. N o tations being as in exam ple 7, le t R= B u R ® r a U M w h ere M = {o?i + a 2 +Q,3 > %a i +a2 }, and le t P = {(e, 1) |o e R ) . L e t g be a n ilp o ten t L ie a lg e b ra of m a x im a l ra n k such th a t P g is eq u iv a len t to P .By p ro p o s itio n 3, g is of type 3 and of c la s s 3; by the d ig re s s io n follow ing exam ple 6, g is iso m o rp h ic to a quo tien t m / a . w h ere m = m (3, 3) and w here C3m £ a. C C 2m.

C o m p arin g P and P m . one can deduce e a s i ly th a t a is of th e fo rm :

a= C v 0 ( m a J w here v is a n o n -z e ro v e c to r in m ai + P2 ° 3\ « * R ~ )

(a is m a n u fa c tu re d in o r d e r to k ill th o se e le m e n ts of R m w hich do not ap p e a r in R).

Now, i t is su ffic ie n t to c o n s id e r th o se a u to m o rp h ism s 0 of in fo r w hich ff(Tm) = T m . w h ere 0 is a s in p ro p o s itio n 1 and w here T m is as in ex am p le 6. Such an a u to m o rp h ism p e rm u te s the b a s is e le m e n ts e x, e2, e 3 w ritte n

in exam ple 7. It follow s th a t two id e a ls as a= C v 0 i and

( Ф a\b = C v ' ® cS Rm— ar e con jugate u n d e r such an a u to m o rp h ism if and onlyV ctèf H— /

IA EA -SM R-11/14 55

i f v and v ' a r e l in e a r ly dependen t (o th e rw ise sa id , th e only p o ss ib le ac tion of the p e rm u ta tio n g roup of th r e e l e t te r s cr3 in P is th e t r iv ia l ac tion ).

H ence n ilp o ten t L ie a lg e b ra s of m a x im a l ra n k having P as sy s te m of w eigh ts a re c la s s if ie d by the com plex p ro je c tiv e sp a ce C P 1.

The l a s t exam ple we w ill g ive h e re is m o re ty p ic a l: a fam ily of a lg e b ra s is c la s s if ie d by a n o n - tr iv ia l quo tien t of a G ra s sm a n n m an ifo ld (in f a c t ,C P 1 again) by a p e rm u ta tio n g roup . The m o s t ty p ic a l exam ple w ould be a fam ily c la s s if ie d by a n o n - tr iv ia l quo tien t of a su b se t of a G ra ssm a n n m an ifo ld by a p e rm u ta tio n g roup , but su ch an exam ple would be m o re d ifficu lt to d e s c r ib e .

E x am p le 9. L e t Ç be a point in the com plex p ro je c tiv e p lane C P 1 , and le t ( z i , Z2 ) be hom ogeneous co o rd in a te s fo r Ç. In th e m odel rn = na(3, 3)

/ н \c o n s id e re d in exam ple 6, le t ae be the su b sp a ce (C (z1e7 - г 2е 8) ) ф ( (J) Ce¡ )

— 4 = 9 '

of rn. As aj is in C 3m , it is an id e a l in m and th e quo tien t = m / a £ is a n ilp o ten t L íe a lg e b ra . Let/t-¡ : n a -* gç be the can o n ica l p ro je c tio n , le tx i = ' / í ( e i ) f ° r i e {1, . . . . 6} and le t x^" be a n o n -z e ro v e c to r in C3 g ç , so th a t {хг, . . . , x7} is a b a s is of g? . N ote th a t, fo r any two po in ts Ç an d -?7 in C P 1, th e two a lg e b ra s g e / C 3g e and gc . /С 3 g e . a r e iso m o rp h ic .

3L e t w= 0 Cx¡ and le t A t(?) be the d e r iv a tio n of g j w hich ex tends the

lin e a r m ap fro m w to g r given by Xj '— ► j q1 o th e rw ise > o r e a °h i = 1, 2, 3

and fo r ea ch Ç e C P 1. (Check th a t th is defin itio n m a k es sense!) As A¡(f) is d iagonal w ith r e s p e c t to the b a s is {xx, . . , x 7}of g £ fo r i= 1, 2, 3, it

3follow s th a t Tg = 0 C A j i s a m a x im a l to ru s fo r g £ , and th a t g £. is of m a x im a l ran k . 1=1

L e t (alta2, a3 ) be the b a s is of T*jj£ w hich is dual to th e b a s is (Д-j, Д2, Д 3) of T ge . The s y s te m of ro o ts can then e a s ily be com puted:

R g£_ = { a 1 . ^ 2 > a 3 > a l + a 2 ’ a 2 + a 3 ’ a 3 + a l > "1 + a 2 + ° 3 ^

and do = dim (g? ) = 1 fo r each a e R g j . H ence the sy s te m of w eigh ts Pgç does not depend on Ç, and w ill be deno ted by P onw ards (F ig . 3).

We w ill now d e s c r ib e a quo tien t sp a ce of C P 1. C o n s id e r C P 1 as the s e t of lin e s in the su b sp a c e E = {(vj, v2 , v 3) e C3 / v j + v 2 + v 3 = 0} of С 3. But then cr3 a c ts in s ta n d a rd fash io n on С 3 by

ct3 X C 3 -----------------------------------► C3

(a, (Vl , v 2, v3 )) '------------------------- ► (v o(1) ,vd2) , v o(3) )

T h is ac tio n in d u ces an ac tio n of cr3 in E, hence in C P 1. The quo tien t sp ace and the can o n ica l p ro je c tio n w ill be deno ted by C P 1 — -— *-СР1/ст3.

A lte rn a tiv e ly , le t m be a point in th is quo tien t sp a ce and le t z = t ' 1 (m ),

56 DE LA HARPE

FIG.3. Picture for P.

w h ere a and b a re two com plex n u m b e rs , each of w hich (but no t both a t thesa m e tim e) m ay be z e ro . Then the o rb it of z u n d e r ct3 is

-i , , I a b b a + b a a + b ] . . . .t (m) = 1 — , — , ------- — , ------ -— , ------- — , ------------f in g en e ra l; th e re a reLb a a + b b a + b a J 6

th re e ex cep tio n a l o rb its w hich a r e , ~ , " lj" , "j“ ~ , - 2, + 1 j- and

2tt\ f . 2-ïïexP v "IT)’ exp v 1 3

T h is being sa id , le t be the s e t o f a ll iso m o rp h ism c la s s e s of n ilpo ten t L ie a lg e b ra s of m a x im a l ra n k w hich have P as sy s te m of w eigh ts. We have c o n s tru c te d a m ap

<ÿ

' g£

It is fa ir ly e a sy to check th a t g j is iso m o rp h ic to gç . w henever Ç and Ç' a r e on the sa m e a 3~ orb it; in the" o th e r w o rd s , the m ap y f a c to rs as

(cj3- o r b i t of Ç)l------------------► g£

U sing m o re re f in e d m e th o d s, G. F a v re h as shown th a t y is ac tu a lly a b ije c tio n . H ence C P 1/ a 3 r e p r e s e n ts the v a r ie ty of n ilp o ten t L ie a lg e b ra s o f m a x im a l ra n k w hich ad m it P as sy s te m of w eigh ts. In p a r t ic u la r , th e re a r e uncountab ly m any iso m o rp h ism c la s s e s in <ÿ.

It can be po in ted out th a t D er(gç ) is so lv ab le if Ç is e i th e r on a g e n e r ic a 3- o rb i t o r on the 2 -p o in ts o rb it , but th a t D e r(g c ) co n ta in s a su b a lg e b ra iso m o rp h ic to £>1(2, C) if Ç is on one of the two s in g u la r 3 -p o in ts o rb its .

A C K N O W L E D G E M E N T

I am g ra te fu l to D. A m iguet, G. F a v re and M. F a v re fo r le ttin g m e consu lt som e o f th e ir unpu b lish ed w ork and fo r s tim u la tin g c o n v e rsa tio n s , as w ell a s to R. C a r te r fo r th e in te r e s t he show ed in th is w ork.

IAEA -SM R -11 /14 57

R E F E R E N C E S

[1] BOURBAKI, N., Groupes et algèbres de Lie, chapitre 1. Hermann (1960).[2] DIEUDONNE, J., Eléments d'analyse 4, Gauthier-Villars (1971). See XIX, section 16, probl. 8 as well

as [1], 4, ex. 18.[3] SAMELSON, H., Notes on Lie algebras, Van Nostrand (1969).[4] AMIGUET, D., Extensions inessentielles d'algèbres de Lie à noyau nilpotent, Thèse, EPFL, Lausanne

(1971).[5] FAVRE, G., Système de poids sur une algèbre de Lie nilpotente, Thèse, EPFL, Lausanne (1972).[6] M O S T O W , G.D., Fully réductible subgroups of algebraic groups. Amer. J. Math. 78 (1956) 200.

M R 19, 1181. _[7] FAVRE, M., Algèbres de Lie complètes, Thèse, EPFL, Lausanne (1972).[8] FAVRE, G., Une algèbre de Lie càractéristiquement nilpotente de dimension 7, C.R. Acad. Sci.

Paris, Série A, 274 (1972) 1338.[9] FAVRE, M . , Algèbres de Lie completes, Thèse, EPFL, Lausanne (1972) 1533.

[10J BOURBAKI, N .. Groupes et algèbres de Lie, chapitres 2 et 3. Hermann (1972).[11] SCORZA, G . , Sulle algebre pseudonulle di ordine massimo, Ann. Mat. Pura Appl. 14 (1936) 1. (I know

of this paper only what is in Zbl. fur Math. 12, 102.)[12] S C H E N K M A N , E., On the derivation algebra and the holomorph of a nilpotent algebra, M e m . Amer.

Math. Soc. 1 4 (1955) 15.M R 16, 993.[13] SATO, T., The derivations of the Lie algebras, Tohoku Math. J. 23 (1971) 21.[14] DYERS, J.L., A nilpotent Lie algebra with nilpotent automorphism group, Bull. Amer. Math. Soc. 76

(1970) 52.M R 40 +2789. _

IA EA -SM R-11/15

ON INFINITE-DIMENSIONAL LIE GROUPS ACTING ON FINITE-DIMENSIONAL MANIFOLDS

P . DE LA H A RPE M athem atics Institute,University of Warwick,Coventry, Warks,United Kingdom

Abstract

ON INFINITE-DIMENSIONAL LIE GROUPS ACTING ON FINITE-DIMENSIONAL MANIFOLDS.The study of groups of diffeomorphisms has m otivated a joined research by H. Omori and the author

about actions of Banach-Lie groups on finite-dim ensional manifolds. The present paper is essentially a list of exam ples which illustrate some results on these actions.

1. INTRODUCTION

A r ic h s o u rc e of p ro b le m s in d if fe re n tia l g e o m e try is p ro v id ed by v a r ia t io n s on the follow ing question :

Q u estio n I. L e t G be a to p o lo g ica l g roup , le t M be a m an ifo ld and

ÍG X M - * Mle t o: i , ■. be an ac tion : a s su m in g th a t M and p s a tis fy c e r ta inH | _(g,m) ^ gm

p r o p e r t ie s , w hat can be sa id abou t G?

We w ill u se th e s ta n d a rd te rm in o lo g y about g roup a c tio n s . In p a r t ic u la r , the ac tio n G X M -» M ist r iv ia l — if gm = m fo r a ll g e G , f or al l m e M ;effec tiv e — if gm = m fo r a ll m e M im p lie s g = e , w h ere e is the

id e n tity of G;tr a n s i t iv e — if fo r any p a i r (m x . m 2 } of po in ts in M, th e re e x is ts g e G

su ch th a t gm ¡ = m 2 .

If G is a connected L ie g roup (o r a B a n a c h -L ie g roup , se e se c tio n 2 below ), a t r a n s i t iv e sm o o th ac tio n G X M -» M is sa id to be

p r im it iv e — if the follow ing holds: le t m Q be so m e po in t in M and le tG 0 = { g e G I g m 0 = m 0} be i ts iso tro p y subg roup ; then theL ie a lg e b ra of G0 is m a x im a l am ong c lo sed su b a lg e b ra s of the L ie a lg e b ra of G.

O bviously , i f the a c tio n is p r im it iv e r e la t iv e to so m e po in t m 0 in M, then the ac tio n is p r im it iv e r e la t iv e to any p o in t in M, b ec au se we have a ssu m e d tr a n s i t iv i ty . About the no tion of p r im it iv ity , s e e D ieudonnè [ 1, (19, p ro b le m s of se c tio n 3)] and R ef. [ 2 ( In tro d u c tio n )] .

59

60 DE LA HARPE

B e fo re m ak ing any p re c is io n about the p a r t ic u la r p h ra s in g of q u es tio n I w hich we w ill c o n s id e r below , we g ive a w ell-know n exam ple ; the f i r s t v e r s io n of it is fo r the r e a d e r s who lik e L ie g ro u p s , and the second fo r th o se who lik e L ie a lg e b ra s .

E x am p le 1. L e t G be a co m p ac t L ie group of d im en sio n к acting sm oo th ly and e ffec tiv e ly on a sm oo th m an ifo ld M of d im en sio n d. T hen k-s d(d + l ) / 2 .

P ro o f: T h is c o n s is ts o f two in e q u a li tie s , fro m w hich the above conc lu s io n fo llow s s tra ig h tfo rw a rd ly .

F i r s t in eq u ality : As the ac tio n is e ffec tiv e , G can be co n s id e re d as a subg roup of the group of d iffe o m o rp h ism s of M. As the g roup is com pac t, it is alw ays p o ss ib le to endow M w ith a R iem an n ian m e tr ic in suc h a way th a t G a c ts by is o m e tr ie s (endow M w ith any R iem an n ian m e tr ic , and then av e ra g e w ith r e s p e c t to the H a a r m e a s u re o f G). L e t now m Q be a po in t of M, le t G0 be the iso tro p y subgroup of m 0 and l e t k 0 be the d im e n sio n of G 0.

As the m ap J о ^ j s in je c tiv e by defin itio n of the iso tro p y[gG 0 K Sm o

subg roup a t m 0, d im (G /G 0 ) S dim(M ) = d.

Second inequality : L e t now T be the tan g en t sp a c e of M a t m 0 , w hich is a E u c lid ea n sp a ce b ec au se M is a R iem an n ian m an ifo ld . F o r each

Гм -» mg £ G 0 le t D(g): T -»■ T be th e d e r iv a tiv e a t m 0 of the m ap gm •

Гм ->■ mAs i s an is o m e try , D(g) is an is o m e try of T , so th a t th is defines

a m ap D: p j g j w h ere O(d) is the o rth o g o n al g roup of T.

As two is o m e tr ie s of M a re id e n tic a l as soon as both th e ir va lue a t som e po in t and th e ir f i r s t d e r iv a tiv e a t th is sa m e p o in t co in c id e (see H elgason [3 (I, le m m a 11. 2) ] , the m ap D is in je c tiv e . As O(d) is of d im en sio n d ( d - l ) / 2 , the in e q u a lity k Q S d(d - 1 )/ 2 ho ld s.

E x am p le l a . L e t G be a connected s im p le com pac t L ie g roup of d im en sio n к ac ting sm oo th ly and n o n - tr iv ia l ly on a sm oo th m an ifo ld M o f d im en sio n d. Then к s d(d + l ) / 2 .

P ro o f: The f i r s t s te p c o n s is ts aga in in p ro v in g th a t к - k Q S d, as in exam ple 1. F o r the second s te p , le t £ be the L ie a lg e b ra of G, le t В: £ X £ -*• R be the K illing fo rm of £ w hich is n eg a tiv e d efin ite (because G is com pac t), le t g^ be th e L ie a lg e b ra of G 0 , and le t m be the o rth o g o n al co m p lem en t of g^ in £ w ith r e s p e c t to B ; as В is n eg a tiv e d efin ite , g is the d i r e c t su m of g^ and of m . F o r any X e g ^ , the l in e a r m ap

adX j^Yh*"fx Y] o rth o g o n a l L ie a lg e b ra defined by the fo rm В

on the v e c to r sp a c e _g; a s is ev id en tly in v a r ia n t by adX, so is i ts

IA EA -SM R-11/15 61

co m p lem en t m . L e t now o(m , B) be the o rth o g o n al L ie a lg e b ra defined by the r e s t r ic t io n of В to the sp a c e m , and le t a be the k e rn e l o f the m ap

is an id e a l in g^ (being the k e rn e l of a h om om orph ism ) and a lso [ a , m ] = {0} (see the d e fin itio n of a ) . H ence a is an id e a l in j [ . As G is connected and as the ac tio n is n o n - tr iv ia l , a canno t be th e w hole of g; but th is im p lie s th a t a = {0}, b ec au se g is s im p le , so th a t the m ap a is in je c tiv e . H ence

w hich c le a r ly co m p le te s th e p ro o f.

R e m a rk s : (1) If G is a co m p ac t s e m i- s im p le group w ith L ie a lg e b ra g,the H a a r m e a s u re on G can be w ritte n s im p ly in te rm s of the K illing fo rm of д . so th a t ex a m p le s 1 and l a a r e indeed v e ry m uch lik e each o th e r .

(2) It shou ld be em p h asized th a t we do need s im p lic ity to apply the p ro o f of ex am p le la , if th a t of exam ple 1 is c o r r e c t fo r any co m p ac t g ro u p .

(3) If G is the g roup SO(n) ac ting in the s ta n d a rd way on the sp h e re then the eq u a lity к = d ( d + l ) / 2 ho ld s.

(4) M uch s tr o n g e r r e s t r ic t io n s on the d im e n sio n of G a r e o ften known, e i th e r w hen one r e s t r i c t s o n e 's a tten tio n to a g iven m an ifo ld , o r w hen one ex c lu d es a s m a ll n u m b e r o f p a r t ic u la r ly " s y m m e tr ic " m an ifo ld s . As two out o f m any p o ss ib le r e f e r e n c e s , we quote [4] and [5] .

2. INFINITE-DIM ENSIONAL LIE GROUPS

L e t M be a co m p ac t sm oo th m an ifo ld (in p a r t ic u la r , M is fin ite d im e n s io n a l) . The s e t o f a ll sm ooth d iffe o m o rp h ism s of M is c le a r ly a g roup , w hich we w ill denote by D iff(M ). T h is g roup c a r r i e s a d iffe re n tia b le s t r u c tu re , w hich h as been in v e s tig a te d by s e v e r a l a u th o rs including L e s l ie [ 6 ] , E b in -M a rsd e n [ 7] , O m o ri [ 8 ] . We w ill not need h e re any deep p r o p e r t ie s of th is s t r u c tu r e , but we w ill only r e c a l l the two follow ing im p o r ta n t fac ts :

(1) The L ie a lg e b ra С (TM) of a l l sm o o th v e c to r f ie ld s on M p la y s in m any r e s p e c ts the ro le of a L ie a lg e b ra fo r the g roup Diff(M ).(2) L o ca lly , Diff(M) looks lik e a F rè c h e t sp a c e . C onsequen tly , the t r a d itio n a l to o ls o f a n a ly s is ( im p lic it function th e o re m , F ro b e n iu s th e o re m ) canno t be app lied w ithout e x tre m e c a re ; indeed , th e se th e o re m s hold in g e n e ra l in th e m o re r e s t r i c t e d co n tex t of B an ach sp a c e s only (see , how ever, s e v e r a l p a p e rs by O m o ri, both p u b lish ed and to ap p e a r) .

A B a n a c h -L ie group is a B an ach m an ifo ld w hich is a g roup and w hich is su c h th a t the group o p e ra tio n s a r e sm o o th . D ealing w ith su ch g ro u p s is m uch e a s ie r th an dea ling w ith d iffe o m o rp h ism g ro u p s , b e c a u se the n a tu ra l

T hen [a, j f ] = [ a , g 0 © m ] = [ a, g Q ] + [ a , m ] c a b ec au se a

k 0 = d im (g0) S d im (o(m , B)) = (k - k 0)(k - k 0 - 1) 2

62 DE LA HARPE

con tex t o f s ta n d a rd a n a ly s is is th a t of B anach sp a c e s (see D ieudonné [9] , in p a r t i c u la r c h a p .X ). A s h o r t in tro d u c tio n to B a n a c h -L ie g ro u p s andth e ir L ie a lg e b ra s can be found in L a z a rd -T i ts [ 10] .

Any fin ite d im e n sio n a l L ie g roup i s obv iously an exam ple of a B anach - L ie group ; o th e rs can e a s i ly be g iven as fo llo w s .

E x am p le 2. L e t H be a s e p a ra b le r e a l o r com plex H ilb e rt sp a ce and le t L(H) be the a lg e b ra of a l l bounded l in e a r o p e ra to rs on H. T hen the group GL(H) of a ll in v e r t ib le e le m e n ts in L(H) is a B a n a c h -L ie g roup , w ith the m an ifo ld s t r u c tu r e being g iven by th e open in c lu s io n of GL(H) in L(H).

The sp a ce L(H) fu rn ish ed w ith the p ro d u c t

f L(H) X L(H) — L(H)J is deno ted by gl(H ); i t is a L ie a lg e b ra and[ (X, Y) и- [X , Y] = XY - YX

a B an ach sp a c e , and the m u ltip lic a tio n is con tinuous; | | [ X , Y] | | S c || X |¡ Y || ; in th is l a s t in e q u a lity , с is a c o n s ta n t w hich can be tak en equa l to 2. An o b je c t su ch as gl(H) is ca lle d a B a n a c h -L ie a lg e b ra .

T he ex p o n en tia l m ap of the B a n a c h -L ie g roup GL(H), w hich is definedex a c tly a s th e ex p o n en tia l m ap of any f in ite d im e n s io n a l L ie g roup , is g iven by the fa m il ia r p o w er s e r ie s

'¿ .(H ) - GL(H)

X H exp X = ^ ¿ X nn = 0

E x am p le 3. S uppose f i r s t th a t H is a f in ite d im e n sio n a l com plex H ilb e r t sp a c e , so th a t GL(H) is d en o ted ,as u su a lly , by G L (n ,^ ) . The follow ing g ro u p s can be defined a s su b g ro u p s of G L(n, <£), in a s im p le way and by a lg e b ra ic equations; U(n), U(p, q) w ith p + q = n, U*(n) w hen n is even, G L (n ,R ). (N otations as in H elgason [3 (IX, 4 ) ] . )

S uppose now th a t H is an in fin ite -d im e n sio n a l com plex H ilb e r t sp a c e . S im ila r c o n s tru c tio n s can be c a r r ie d o v e r to th is ca se ; fo r ex am p le , the u n ita ry g roup U(H) of H is defined by (V e G L (H ) | VV* = V*V = 1}; i t is e a sy to ch eck th a t U(H) is a B a n a c h -L ie g roup w ith L ie a lg e b ra u(H) = { X e g l(H ) I X* = -X } .

A s y s te m a tic s tudy of su ch B a n a c h -L ie g ro u p s has been s ta r te d , s e e R ef. [ 11 ] .

A second a p p ro x im a tio n to the q u es tio n of in te r e s t h e re ca n now be fo rm u la ted :

Q u estio n II. L e t M be a f in ite d im e n sio n a l m anifo ld ; is i t p o ss ib le to find an in fin ite d im e n s io n a l B a n a c h -L ie g roup G and an ac tio n of G on M sa tis fy in g c e r ta in p re im p o se d cond itions?

Any e ffec tiv e ac tio n of a g roup G on M defines obviously an in je c tio n o f G in Diff(M ). H ence y e t an o th e r p h ras in g :

IA EA -SM R -11/15 63

Q u es tio n l ia . If M is as above, w hat a r e the su b g ro u p s G of Diff(M) w hich can be fu rn ish e d w ith a B a n a c h -L ie group s t r u c tu re su ch th a t the ac tio n G X M -» M induced by the ev a lu a tio n m ap Diff(M) X M - * M i s sm ooth?

A bout the re la tio n sh ip be tw een th is q u es tio n and o th e r c o n s id e ra tio n s on group Diff(M ), r e m e m b e r P ro f e s s o r E e l l s ' le c tu re in th is S u m m er C o llege ("T he d iffe o m o rp h ism group in a n a ly s is " ) , w hich i s no t p u b lish ed in th e se P ro c e e d in g s .

3. B A N A CH -LIE GROUPS WHICH DO NOT ACT ON FINITE-DIM ENSIONAL MANIFOLDS

L e t G be a r e a l B a n a c h -L ie g roup ac ting sm o o th ly on a m an ifo ld M of fin ite d im e n sio n d. L e t m 0 be a p o in t o f M, and le t G 0 , £ and gQ be defined a s in se c tio n 1. C hoose an e le m e n t X £ | , and le t

IR GJ be the c o rre sp o n d in g o n e -p a ra m e te r subg roup of G. F o rjtH - exp(tX)

deach p o in t m in M, th e q u an tity ^ ((exp(tX))m ) is a tan g en t v e c to r a tt = o

m to M; hence one has a v e c to r f ie ld on M w hich w ill be denoted by т и - X (m ). E v e ry th in g being sm oo th , th is v e c to r f ie ld has a T a y lo r s e r ie s X a t m 0.

L e t now & be the sp a ce o f " fo rm a l p o w er s e r i e s o f lo c a l v e c to r f ie ld s a t the o r ig in of IRd", n am ely the sp a c e of a ll e x p re s s io n s of the fo rm

I Ii = l a £ N a

A„ ЭЭх1

w h ere the A ^ 's a r e r e a l c o n s ta n ts and w h ere we have used the tra d i t io n a l n o ta tio n fo r m u lt i- in d ic e s . The sp a c e J M s m ade a r e a l L ie a lg e b ra in the obvious w ay, and the c o n s tru c tio n above d efin es a h o m o m o rp h ism

Ф -g - S * -

Хи- X

F o r each k e { -1 , 0, 1, 2, . . . } , define th e su b sp ace

d

ylp' I °,s"Jv = 1 p e , , ,i = l a 6 N d

l a lak+ l

64 DE LA HARPE

T hen the fam ily (Jk)ke { -i, o, ..} *s a f i l t r a t io n of the L ie a lg e b ra â*. This m e a n s th a t

(1) J - i 3 J 0 э J x э . . . d J k d J k+1 э . . .

w ith J k = & and J k = {0}

к = -1 k = -l

(2) [ J k , J ( ] C J k t ( fo r a l l k , i e { - l , 0, 1, . . . }

(3) d im (Jk / J k + 1 ) < oo for a l l к е { - 1 , 0, 1, . . . }

M o re o v e r , the r e su l t in g f i l t e r e d L ie a lg e b ra is t r a n s i t iv e :

(4) fo r ev e ry k e { 0 , 1, 2, . . . } , fo r ev e ry P e J k

such th a t P ф Jk+i • th e r e e x i s t s XeS*"

such tha t [X, P] ф J k

(That condition 4 i s a t r a n s i t iv i ty r e q u i r e m e n t i s b e s t s e e n by looking at D ieudonnê [ 1 (p rob lem 5 d of se c t io n 1 9 .3 ) ] .

Defining now gj = Ф 1(Jk) fo r a l l k e { - l , 0, 1, . . .} , we ob ta in a f i l t r a t io n (ёк)ке{-1 o } °^ B a n a c h -L ie a lg e b ra g_. (See r e m a r k ju s t befo re

exam ple 5 fo r the condition O i i = i ° b )к =-1

The study of (poss ib ly in fin ite d im ensiona l) f i l t e r e d L ie a lg e b ra s , in i t ia ted by E . C a r t a n [ 12], has p ro d u ce d a v e r y r i c h and deep theory ; a s an e le m e n ta ry a p p l i c a t io n o f i t and of c l a s s i f i c a t io n r e s u l t s [ 16], the following two th e o re m s have been p ro v ed in Ref. [ 13] :

T h e o r e m 1. L e t G be a connected B a n a c h -L ie group , which i s second- countab le , and which a c ts sm ooth ly and effec tive ly on a f in ite d im e n s io n a l m an ifo ld M. If the ac tion i s p r im i t iv e , then G m u s t be f in ite d im ens iona l .

T h e o r e m 2. L e t g be the L ie a lg e b ra of an infin ite d im e n s io n a l B a n a c h -L ieg roup G. Suppose g has no p r o p e r c lo sed fin ite co d im en s io n a l idea l . T hen jf has no p r o p e r c lo sed fin ite co d im en s io n a l su b a lg eb ra . In p a r t i c u la r , any sm ooth ac t io n of G on a f in i te -d im e n s io n a l m anifo ld is t r iv ia l .

The d e ta i led p ro o fs cannot be g iven h e r e . T h e i r p r in c ip le i s , how ever, quite s im p le , and i s b e s t exposed by the following.

E x am p le 4. L e t h be the L ie a lg e b ra of v e c to r f ie lds on R g e n e ra te d by the e le m e n ts

Э Э 2 _Э_ n ЭЭх ’ X Эх ' X Эх ' ‘ ‘ ’ X Эх ’

Then th e re i s no B a n a c h -L ie a l g e b r a £ such tha t h is ( isom orph ic to) a su b a lg e b ra of g.

lA EA -SM R -11/15 65

P roof : Suppose indeed th a t th e r e e x i s t s a n o rm || || on h and a co n s tan t с su c h th a t || [ A, B] || é с || A j| || В || fo r a l l A, B e h. Then

U П U

X Эх ’ X Эх S с Эх

for a l l in t e g e r n, which i s c l e a r ly a b s u rd .Hence the a lg e b r a Ф(£) (notations as b e fo re th e o re m I) cannot be

is o m o rp h ic to the a lg e b r a h of exam ple 4, b ec au se 'í'(g) i s a B a n a c h -L ie a lg e b ra . U sing the known c l a s s i f i c a t io n of r e a l " p r im i t iv e f i l t e re d L ie a lg e b r a s " , one can exclude a l l in fin ite d im e n s io n a l c a s e s by s im i l a r com pu ta t ions ; hence th e o re m I and, w ith h a rd ly any m o r e work, th e o re m II.

R e m a rk : In supposing above th a t Ф(|[) i s a B a n a c h -L ie a lg eb ra , we have im p l ic i t ly a s su m e d th a t the m ap Ф is in jec t ive ; th is is a lways the c a s e i f the ac t io n of G on M is ana ly tic ; though th e o re m s I and II a r e t ru e fo r sm o o th ac t ions in g e n e ra l , we cannot e n te r into d e ta i l s h e r e and we r e f e r to R ef. [ 13] .

E x a m p le 5. L e t GL(H) be a s in exa m p le 2, w ith H a r e a l inf in ite d im e n s io n a l H i lb e r t sp a c e . Then any ac tion of GL(H) on a fin ite d im e n s io n a l m an ifo ld is t r iv ia l .

P roo f : The id e a l s t r u c t u r e of the a s so c ia t iv e a lg e b ra L(H) i s w ell- known (see fo r exam ple S chat ten [ 14 (I. 6)] . U sing r e s u l t s on the L ie s t r u c t u r e of an a s s o c ia t iv e a lg e b ra as in B ourbak i [ 15 (1, ex. 7)], one can e a s i ly check th a t any p r o p e r id e a l o f ¿1(H) I s contained in the idea l

so th a t any p r o p e r id e a l of gl(H) is a f o r t i o r i of in f in ite cod im ens ion .Hence th e o re m II app l ies and ends the p roof. The conc lu s ion could n a tu ra l ly be expected a f te r exam ple l a .

4. A BANACH-LIE GROUP WHICH ACTS E F F E C T IV E L Y AND TRANSITIVELY ON R 2

The hypo thes is of p r im i t iv i ty i s e s s e n t i a l in th e o re m I, as w ill be shown in th is se c t io n .

L e t E be the sp a c e of th o se an a ly t ic r e a l functions of one r e a l v a r ia b le whose T a y lo r s e r i e s at the o r ig in is of the fo rm

v e rg e n c e of any such s e r i e s i s in f in ite . ) Then E is a r e a l B anach sp a ce for the n o rm defined by ||f | | = s u p | a nn! | .

n £ N

F o r ea ch s € R and fo r ea ch f e E , l e t fs be the function x k f(x + s) .

X a r e a l n u m b e r

Y a co m p a c t o p e r a to r on H

sup I a nnl I <oo. (In p a r t i c u la r , the r a d iu s of con- n£Nn = 0

66 DE LA HARPE

L e m m a 1. F o r each s e IR and f e E , f s is in E; m o r e o v e r , the

ftR X E - E m apping r):4 is sm ooth .

[(s,f)H- fs

Sketch of p roof:oo 00 n

Step one: If f(x) anx n, then fs (x) x ks n_k =n = 0 n = 0 k = 0

00 00

=I Œa" ( n - k ) i ki) x k - Hence iifsii- sup(I |s,n’k) s | | f | |e 'slk=0 n=k k€N n=k

w hich p r o v e s the f i r s t a s s e r t io n .

Step two: F o r ea ch in te g e r m , le t f (m> denote the m - th d e r iv a t iv e of f. I t i s t r i v i a l to ch eck th a t f^m' e E and th a t || f (m || s || f || . F o r each k e N and fo r ea c h s e lR , l e t now <pk(f, s) be the function

К

f(x + s) - ^ - ¿ 7 f (m) (x)sm

it i s obvious th a t (pk(f, s) belongs to E; d i r e c t com pu ta t ions using T a y lo r ' s fo rm u la show, m o r e o v e r , tha t

II <Pk(f, t ) II s | t |k+1 e N ||f||and tha t

Ml M ' l,M"fo r each r e a l n u m b e r t.

F o r ea ch g £ E and s , t € R , l e t now F k (f, g; s , t ) be the function

к k-1

x K (f + g)(x + s + t ) - j T ^ f (m) (x + s ) tm + ^ g (m,(x + s ) tm|m=0 m=0

2 и 2and le t 6 be the n u m b e r J | t | + | |g || . Then, i f 6 f 0,

k¿ Fk (f,g ; s , t ) (x) j f t x + s + t) f<m> (x + s) f

m=0

+ 7E 1 g ( x + s +

k-1

t) g(m) (X + S) T

IA EA -SM R -11/15 67

so th a t 11^-Fk (f,g; s , t ) || S p p r I K (fs , t) || + ||<Pk . i (gs , t) ||

s | t | e |s| +lt| | |f | | + e |s| +lt| Il g II

Hence l im Ц- i Fk (f, g; s , t ) | | = 0.6 -> о

Step th re e : F o r each k e N and a s su m in g it e x i s t s , the к - th d e r iv a t iv e of the m ap r¡ a t (s, f) is denoted by (Dk rj) ( s ,f ) and is a s y m m e t r i c m u l t i l in e a r m ap

(R ® E) X ..........X (R ® E) - E

к t im e s

We c la im th a t th is d e r iv a t iv e e x i s ts and is g iven as follows:(Dkr)) (ti> gj ) ( t2> g2 ) • • • ( t k. g k) is the function which ta k e s a t x the value

к

f (k) ( x + s H - lV , t k + ^ gf 4 (x + s H j . . t j . . t k j=l

In o r d e r to p ro v e th is c la im , i t i s su f f ic ien t to p ro v e th a t (Dk rj) (t, g) . . . . (t, g) i s the function which ta k e s a t x the va lue <s,f) --------- »------------ '

к t im e sf (k) ( x + s ) t k + k g ^ ' ^ (x + s ) t k " 1

and th is i s p r e c i s e l y what has been p roved in s te p two. Hence the le m m a .

L e t now G be the B a n a c h -L ie g roup whose under ly ing m anifo ld is the B a n ac h s p a c e R Ф E and w hose p ro d u c t is defined by (s ,f )( t , g) = (s + t, ft +g). L e m m a 1 shows th a t the group o p e ra t io n s a r e sm ooth , so th a t G is indeed a B a n a c h -L ie group .

Í E X R - RL e m m a 2. L e t f be the eva lua tion m ap -j . T hen f is

sm oo th . (f, x) и- f(x )

kP ro o f . F o r ea ch k e N , it is e a sy to check th a t (D f) is the

s y m m e t r i c m u l t i l in e a r m ap defined by ^

(° k ç)(f,x) t e r M b . y2 >■ • • tek • yk ) = f(k) (*&! y2 • ■ • ykк

+ Z i"4 (Х)У1 "•У! •••УкHence the le m m a . j = i

2L e t now G be ac ting on R by

(G X R2 R2

((s, f), (x, у)) и. (s + x , f(x) +y)

. L e m m a 2 shows that this action i s smooth.

68 DE LA HARPE

It i s t r i v i a l to check tha t th is ac tion is e ffec tive and t r a n s i t iv e . The2 i i so t ro p y subgroup of G a t the o r ig in of R is G0 = {(s, f ) £ G | s = 0 andf(0) = 0}. In a g r e e m e n t with th e o re m I, th is ac tion is not p r im i t iv e ;indeed: G j C E c G , the cod im en s io n of G0 in E is +1 and the cod im ens ionof E in G is +1. (E is n a tu ra l ly iden tif ied with the subgroup { ( s , f ) e G | s = 0}of G . )

G e o m e tr ic a l ly speak ing , i t should be no ted th a t the ac t io n d e s c r ib e d in th is s e c t io n has a c e r t a in " s y m m e t ry " (this i s not a tech n ic a l ly w ell- defined t e r m , at l e a s t not a t p r e s e n t ) which can be po in ted out as follows.L e t (s, f) be a f ixed e le m e n t of G; a t ea ch point m of R 2 , le t v(m) be the v e c to r (s, f)m - m; then the v e c to r f ield v on R2 i s in v a r i a n t by v e r t i c a l t r a n s l a t i o n s . I t is in s t ru c t iv e to d raw p ic tu r e s , say when the e le m e n t of G is (0, f) o r ( i , f) and when f is the function defined by f(x) = x.

5. CONCLUDING REMARKS

The m o r a l to r e ta in of th e o re m I is tha t the study of B a n a c h -L ie g roups and th a t o f g ro u p s of d i f fe o m o rp h ism s a r e in fac t v e r y d if fe ren t su b je c ts of in v e s t ig a t io n s , even though both a r e typ ica l ly inf in ite d im ens iona l .

The exam ple of s e c t io n 4 su g g e s ts th a t ac tions of (possib ly in fin ite d im ens iona l) B a n a c h -L ie g roups on fin ite d im e n s io n a l m an ifo lds a r i s e in connect ion with fa m i l ie s of v e c to r f ie lds p o se s s in g c e r t a in " s y m m e t r i e s " .

R E F E R E N C E S

[1 ] DIEUDONNE, J . , Elements d'analyse 4, Gauthier-Villars (1971).[2 ] GUILLEM1N, V ., Infinite dimensional prim itive Lie algebras, J. D ifferential Geometry 4 (1970) 257.

MR 42 +3132.[3 ] HELGASON, S . , D ifferential Geometry and Symm etric Spaces, A cadem ic Press (1962).[4 ] HSIANG, Wu-Yi, On the bound of the dimensions of isometry groups of all possible riem annian metrics

on an exotic sphere, Ann. of M ath. 8^(1967) 351. MR 35 +4935.[5 ] KOBAYASHI, S . , Transformation Groups in D ifferential Geometry, Springer (1972).[ 6] LESLIE, J. A . , On a differential structure of the group of diffeomorphisms, Topology 6 (1967) 263.

MR 35 +1041.[7 ] EBIN, D. G . , M ARSDEN, j . , Group of diffeomorphisms and the m otion of an incompressible fluid,

Ann. of M ath. 92 (1970) 102. MR 42 +6865.[ 8] OMORI, H . , Local structures of groups of diffeomorphisms, J. M ath. Soc. Japan 24 (1972) 60.[9 ] DIEUDONNE, J . , Les fondements de l'analyse moderne, Gauthier-Villars (1963) or A cadem ic Press (1960).

[10] LAZARD, M ., TITS, J . , Domaines d 'in jec tiv ité de l ’application exponentielle, Topology 4 (1965/66) 315. MR 32 +3518.

[11 ] DE LA HARPE, P . , C lassical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space.(To appear as M athem atics Lecture Notes 285, Springer.)

[12] CARTAN, E ., O uvres com plètes, (Part II, v o l.2) Gauthier-Villars (1953).[ 13] OMORI, H . , DE LA HARPE, P . , About interaction between Banach-Lie groups and finite dimensional

manifolds, J. Math. Kyoto Univ. 12 (1972) 573. MR£7_. 4286.[14] SCHATTEN, R ., Norm Ideals of Com pletely Continuous Operators. Springer (1960).[15] BOURBAKI, N ., Groupes e t algèbres de Lie, chapitre I, Hermann (1960).

IA EA -SM R -11/15

A D D I T I O N A L B I B L I O G R A P H Y

F r o m the v a s t l i t e r a t u r e c o n c e rn e d with the c l a s s i f i c a t io n r e s u l t s used to p ro v e th e o re m s I and II, one can quote b e s id e s R efs [ 2] and [ 12] the four following:

KOBAYASHI, S . , NAGANO, T . , On filtered Lie algebras and geom etric structures I-V , J. M ath. M ech. (1964-1966). M R29 + 5961, 32 +2512 and 5803, 33 +4188 and 4189.

SINGER, I .M . , STERNBERG, S . , The infinite groups of Lie and Cartan, part I (the transitive groups),J. d 'an . m ath. 15 (1965) 1. MR 36+911.

MOR1MOTO, T . , TANAKA, N ., The classification of the real prim itive infinite Lie algebras, J. M ath. Kyoto University 10 (1970) 207. MR 42 +3133.

SCHNIDER, S ., The classification of real prim itive infinite Lie algebras, J. Diff. Geom. 4 (1970) 81.

On the notion of a B a n a c h -L ie group as in troduced in s e c t io n 2, s e e the r e c e n t ly pub lished G roupes e t a lg è b r e s de L ie , chap. 3, by B ourbak i (H erm ann, 1972).

IAEA-SM R-11/1 6

SOME PROPERTIES OF INFINITEDIMENSIONAL ORTHOGONAL GROUPS

P. DE LA HARPE M athem atics Institute,University of Warwick,Coventry, Warks,U nited K ingdom

Abstract

SOME PROPERTIES OF INFINITE-DIMENSIONAL ORTHOGONAL GROUPS.The standard notion of a (real or complex) Lie group can be extended in many ways. One of them

leads to the study of the so-called Banach-Lie groups and of their Lie algebras. An im portant part of the theory of finite-dim ensional Lie groups can easily be seen to carry over to the Banach case; but several interesting properties characteristic of infinite dimensions have recently been (or are now being) discovered.One of them is discussed in this paper.

Section 1 recalls properties of classes of com pact operators acting on a Hilbert space H (the von N eum ann-Schatten ideals). Section 2 contains the definitions of the groups of interest here, which are groups of orthogonal operators on H. Section 3 shows how to compute the homotopy types of these groups; section 4 very briefly indicates a connection with the theory of representations of the canonical anticom m utation relations of quantum physics.

The idea of section 3 is due to H. Pittie.

1. REVIEW O F OPER ATOR THEORY(VON NEUM ANN-SCHATTEN IDEALS)

L e t H be a r e a l H i lb e r t sp a c e of an in fin ite n u m b e r of d im e n s io n s . The B anach a lg e b ra of a l l continuous l in e a r o p e r a to r s on H w ill be denoted by L(H), and tha t of a l l co m p ac t o p e r a to r s on H by C(H). If X i s in C(H), i t i s c l a s s i c a l tha t the s p e c t r u m of X (by which we m e a n the u su a l s p e c t r u m of the o p e r a to r X Œ which ex tends X l in e a r ly on the com plex if ica t ion H Œ of H) con ta ins at m o s t countably m any po in ts , which a re a l l iso la te d e igenva lues (of X a-) with the only p o ss ib le exception of 0. The o p e r a to r XX* (where X* i s the ad jo in t of X) has a unique p os it ive s q u a re ro o t , which i s aga in a c o m p ac t o p e r a to r , denoted by [X]. The s p e c t r u m of [X] c o n s is t s of, at m o s t , countably m any iso la te d s t r i c t ly pos i t ive n u m b e rs and of 0. We sh a l l denote by (nDQi))n = i,2,... an e n u m e ra t io n of the e igenva lues of [X], r e p e a te d ac c o rd in g to th e i r m u lt ip l ic i ty , and o r d e r e d such tha t

Ml(X) È M2(X) è .......... ё м„(Х) ё ..........

If the s p e c t r u m of [X] i s f in ite , the n u m b e rs /jn(X) 's a r e a l l 0 fo r n l a rg e enough. In any c a s e , the se quence (/un(X)) tends to w ard s z e ro .

In a l l of w hat fo llow s, p i s e i th e r a r e a l n u m b e r l a r g e r than one, o r the sy m b o l + oo :

р е И , 1 È p g + oo

71

72 DE LA HARPE

L et us now define the von N eu m a n n -Schatten id e a ls Cp (H):

П = 1fu rn ish ed with the n o rm || ||p defined by

n = 1

llxllp = ( V ( M n(X))] )1/p

,p

I f p = c o : C«(H) = C(H) fu rn ish ed with the n o rm || || (or || || „ ) inducedf ro m the B anach a lg e b ra L(H).

We r e c a l l , in a f i r s t p ro p o s i t io n , so m e p r o p e r t i e s of th e se :

P ro p o s i t io n 1

(i) Cp(H) i s an id e a l in L(H).(ii) Cp(H) is a r e a l involutive Banach a lg eb ra .

(iii) If X i s in Cp(H) and if U, V a r e o r thogona l o p e r a to r s on H, thenIIUXV ||p = IIX ||p .

(iv) If X i s in Cp(H) and i f A, В a r e in L(H), then | | a X b | p S | | a | | ||х ||р ||в||.(v) If 1 s p s p ' s » , then the in jec t ion Cp(H)->- C p'(H) i s continuous.

(vi) L e t (en ) j=i 2 be an o r th o n o rm a l b a s is of H; for ea ch n, le t Mn bethe se t of those o p e r a to r s which m ap the span of { e b . . . , e n} intoi t s e l f and i t s o r thogona l co m p lem en t onto z e ro ; le t M„ be the unionof a l l the M n 's . Then M„ is d en se in C p(H); in p a r t i c u l a r , the s e t of f in i t e - r a n k o p e r a to r s on H i s d en se in C p(H).

D eta iled p ro o fs a r e given, e. g. by Schatten [12]. The m o s t in te r e s t i n g c a s e s in a n a ly s is a r e p = l (n u c le a r , o r t r a c e - c l a s s , o p e r a to r s ) , p = 2 (H i lb e r t-S c h m id t o p e r a to r s ) and p = °o (com pac t o p e r a to r s ) ; typ ica l exam ples can a lso be found in R e f . [2 ] , s e c t io n 1 5 .4 , p ro b le m 14, and s e c t io n 15 .11 , p ro b le m 7.

2. GENERAL LINEAR GROUPS AND ORTHOGONAL GROUPSO F OPERATORS ON H

The space L(H) can be viewed as a ( re a l ) B a n a c h -L ie a lg e b ra , with the p ro d u c t defined by [X,Y] = X Y -Y X , and w ill in th is c a s e be denoted by _gl(H). Then j[l(H) is the L ie a lg e b ra of the B a n a c h -L ie g roup GL(H) of al l in v e r t ib le o p e r a to r s on H. Some m o r e d e ta i l s a r e given in the second p a r t of s e c t io n 2 of Ref. [6].

L e t o(H) be the sub B a n a c h -L ie a lg e b r a of a l l sk ew -ad jo in t o p e r a to r s in gl(H). It i s ea sy to check th a t £(H) i s the Lie a lg e b ra of the B a n ac h -L ie g roup O(H) c o n s is t in g of a l l o r thogona l o p e r a to r s on H.

S im ila r ly , the sp a ce C p(H) can be viewed a s a B a n a c h -L ie a lg e b ra which w ill be denoted by gl(H; Cp ). L e t now

GL(H; Cp) = { T e G L ( H ) | T - I e C p(H)}

IAEA-SM R-11 /16 73

w h ere I i s the iden ti ty o p e r a to r on H. It i s a group ( indeed , a n o r m a l subg roup of GL(H)) b ecause C p(H) i s an id e a l in L(H). The m ap

i G L ( H ; C p ) ------------ - C p ( H )

[ T I--------- - T - I

i s a b isection of GL(H; C p) onto the open su b se t of Cp(H) conta in ing those o p e r a to r s for which +1 i s not an e igenva lue . T h is endows GL(H; Cp ) with the s t r u c t u r e of a B anach m anifo ld . F in a l ly , the group o p e ra t io n s a r e sm oo th with r e s p e c t to th is s t r u c t u r e , b e c a u s e the m u lt ip l ic a t io n in Cp (H) is continuous (hence ana ly tic ; s e e Ref. [1], exam ple 8 .1 2 .9 ) . In o th e r w ords GL(H; Cp) i s a B a n a c h -L ie group; i t i s v e ry ea sy to check tha t gl(H; C p) i s i t s L ie a lg e b ra and tha t the exponen tia l m ap i s given by the t r a d i t io n a l pow er s e r i e s .

F in a l ly , le t o(H; Cp ) be the B a n a c h -L ie a lg e b ra of a l l sk e w -ad jo in t o p e r a to r s in gl(H; C p). T h is i s the L ie a lg e b ra of the B a n a c h -L ie group

0(H ; Cp ) = (T e O (H ) | Т - i e C p(H)}

= {T6G L (H ; С р) | т т * = T*T = 1}

It i s an ea sy and in s t ru c t iv e e x e r c i s e to t r a n s l a t e fo r G1(H; C p) and 0(H; C p ), se c t io n 7 .2 of L ang [8], who c o n s id e r s G1(HŒ) and U(HŒ ), th e i r exponen tia l m a p s and the p o la r deco m p o s i t io n of in v e r t ib le o p e r a to r s .

Note th a t the function j ° ^ H ' c p)x ^ H , Cp)^ >- R+^ def ines

a b i - in v a r ia n t d is ta n c e on the group 0 (H ; C p); the a s s o c ia te d topology is th a t un d e r ly in g the m an ifo ld s t r u c t u r e of 0(H; C p) defined above. O ur t a r g e t i s to study the homotopy type of th e se topo log ica l g ro u p s , in t e r m s of the homotopy types of the ( f in i te -d im e n s io n a l) c l a s s i c a l L ie groups O(n).

3. HOMOTOPY TYPES

It follows t r iv ia l ly f ro m p o la r d ec om pos i t ion ( see Lang [8], c h a p te r 7, p ro p o s i t io n 6) th a t G1(H) and O(H) have the s a m e homotopy type , and the s a m e holds for GL(H; Cp ) and 0 (H ; C p). H ence , i t i s su ff ic ien t to c o n s id e r o r thogona l g roups only in th is sec tion .

The m o s t fam ous r e s u l t i s K u ip e r ' s th e o re m [7]:

P ro p o s i t io n 2 . The B a n a c h -L ie group O(H) i s c o n t ra c t ib le . The p ro o f of p ro p o s i t io n 2, o r even a sk e tch , i s beyond the scope of th is le c tu r e .

The homotopy type of the g roups 0 (H ; C p) is m uch e a s i e r to com pute and has been w orked out (e s se n t ia l ly independently) by E lw orthy , Geba, P a la i s , S v a rc , T ro m b a ( see the r e f e r e n c e s given in Ref. [4]). The m ethod of p ro o f of p ro p o s i t io n 3 below h a s , to o u r knowledge, not ye t been published . We have l e a r n e d i t f ro m H. P i t t ie .

74 DE LA HARPE

L et ( e n)n = i i 2l . . . b e an o r th o n o rm a l b a s is of H. F o r ea ch n, we identify the E uc l idean sp a ce Rn to the span of {e j, . . . , e n} in H and the co m p ac t L ie g roup O(n) to the subg roup of 0(H; Cp ) c o n s is t in g of th o se o p e r a to r s which m a p R n onto i t s e l f and whose r e s t r i c t i o n to the o r thogona l co m p lem en t of R n in H i s the iden ti ty (O(n) = 0(H; C p)DM n , w here M n i s a s in p ropos it ion l(v i) ) . T h ese id en t i f ica t ions give r i s e to the s ta n d a rd in c lu s io n s

0 (n )C 0 (n +1 ) C . . .C O ( c o ) ---- }-----► 0(H; Cp)

r 00w h ere O(°o) is the topo log ica l group O(n), nam ely the group U O(n)fu rn ish ed with the inductive l im i t topology. n_1

P ro p o s i t io n 3. The in c lu s io n O(oo)---- — >-0(H; C p) i s a homotopy equ iva lence .

P ro p o s i t io n 3 r e d u c e s the com puta t ion of the homotopy type of 0(H; Cp ) to th a t of the s ta b le o r thogona l g roup , which is known (see , e .g . M ilnor [10], p a r t 4). F i r s t , we p rove th r e e le m m a s :

L e m m a 1. L e t T0 and Tj be two e le m e n ts in 0 (H ; Cp ) such tha t ||Tj - T 0 ||p < l . T hen T 0 and Tj can be jo ined by a continuous path in 0(H; C p ).

P ro o f . By p ro p o s i t io n 1(iii) , H t ' ^ T j - T q ) ! s || T<J1 || || T x - T0 ||p < 1, so tha t the pow er s e r i e s w hich g ives log(I + T51( T 1 - T 0)) co n v e rg es abso lu te ly and

def ines an e le m e n t , say S, in C n(H). The m a p { I t 1 *■ l j - T q exp(tb)

def ines now a continuous path connect ing T0 with Tj .

As an im m e d ia te c o r o l l a r y , if X is a topo log ica l sp a ce and i f f 0, f j a r e two continuous m a p s f ro m X to 0 (H ; Cp ) such th a t sup | |f i (x ) - f0(x)|| < 1, then f0 and f j a r e hom otop ica lly equivalen t. xeX

L e m m a 2. L e t X be a co m p ac t po lyhedron and le t f: X — ► 0(H; Cp) be acontinuous m ap . Then th e re e x i s t s an in te g e r m and a m ap g: X ---- ►O(m)such th a t f and g a r e hom otop ica lly equ iva len t in 0 (H ; C p).

P roo f . F o r each in te g e r n, c o n s id e r O(n) with i t s s ta n d a rd R iem ann ian s t r u c t u r e ( se e , e. g. M ilnor [10], p a r t 4). L e t e be a s t r ic t ly pos i t ive r e a l n u m b e r , s m a l l e r than 1 /5 , and sm aU enough fo r a l l the ba l ls of r a d iu s 3 e in O(n) to be convex. (This m e a n s th a t , given any two poin ts in s ide such a ba l l , th e r e e x i s t s a unique geodesic se g m en t in s ide th is ball which goes f ro m one point to the o th e r ; i t is a fac t th a t the in te r s e c t io n of two of th e se balls i s aga in convex; such an e can be chosen independently of n. )

L e t d be a d is ta n c e on X com patib le with the topology. As X is com pac t, f i s un ifo rm ly continuous and th e re e x i s t s a s t r i c t ly pos i t ive r e a l nu m b e r e' such th a t , fo r a l l x, y G X with d(x, y) < e 1, the inequali ty || f(x) - f(y ) ||p < e ho lds. L e t К be a t r ia n g u la t io n of X such th a t ea ch s im p lex о of К i s con ta ined in s id e an open ba l l of r a d iu s e 1 in X; the v e r t i c e s of К w ill be denoted by Xj, . . . , x k. As O(oo) i s d en se in 0 (H ; C p) by p ro p o s i t io n l(v i) , fo r ea ch i e { l , . . . , k} th e r e e x i s t s an in te g e r m¡ such tha t the d is tan c e between f(x¡ ) and 0 ( m ;) i s s m a l l e r than e. L e t m be the l a r g e s t of the m ¡ 's , Then, for any x e X , the d is tan c e between f(x) and O(m) i s s m a l l e r than 3 e.

IA EA -SM R-11/16 75

F o r each i e { l , . . . , k} , le t y¡ be a point in O(m) such th a t || f(x¡) - y¡ ||p < e.Define a m a p g: X ---- ► 0(H ; C p) as fo llows: On the v e r t i c e s of K, putg(x¡) = y ¡ . L e t now a be a c lo sed s im p le x in К which i s contained in no o th e r c lo sed s im p le x than i t s e l f (one of the " l a r g e s t " s im p l ic e s ) ; if x j , . . . , Xj(0j a r e the v e r t i c e s of a , then î(xi), . . . , f (x j(c)) a r e contained in so m e convex ba l l , say B 0, of r a d iu s e in O(m); hence , yi , . . . , yj(0) a r e conta ined in som e convex ball BJ, of r a d iu s 3 e in O(m); extend g to m ap continuously a in B¿,, w ith the following condition: if so m e su b s im p le x of ó i s a l so a s u b s im p le x of an o th e r of the " l a r g e s t " s im p l ic e s , say f, then the c o r r e s ponding y j ' s m u s t a lso be contained in the co r r e sp o n d in g ba l l В | of r a d iu s 3 e; ( th e re i s no o b s t ru c t io n to th is ex tens ion of g b ecause a l l the convex ba l ls and th e i r in te r s e c t io n s a r e co n t ra c t ib le ) . T h is p ro c e d u re can be r e p e a te d for a l l the c lo sed s im p l ic e s of К a s above, one a f te r the o th e r , tak ing ea ch t im e c a r e of those s u b s im p l ic e s w here the m ap m ay a l re a d yhave been defined. One o b ta in s , in th is way, a continuous m ap g: X ---- ►O(m)such th a t sup j|f(x) - g(x)|| < 5 e. Hence, by le m m a 1, f and g a r e hom o-

x e Xtop ica l ly equivalen t .

L e m m a 3. L e t X be a co m p ac t po lyhedron and le t f0 , f j : X-----»-0(oo) betwo continuous m a p s which a r e hom otop ica lly equ iva len t in 0 (H ; Cp ). Then f Q and fj a r e hom otop ica lly equ iva len t in O(oo).

Idea of p ro o f . L e t F : X x [0 ,1 ] ----»-0(H; Cp ) be a homotopy of f0 to w a rd sf j in 0 (H ; Cp ). L e t L be an ad hoc t r ia n g u la t io n of X x [ 0 , 1] which contains a su b tr ia n g u la t io n К of (X x{0})U (Xx{l}). By h y p o th e s is , F m aps К in the subgroup O(oo) of 0 (H ; Cp). P ro c e e d in g v e ry m u c h l ike in the p ro o f ofl e m m a 2, i t i s p o ss ib le to define a m ap G: Xx [0, 1 ] ----►O(oo) which isc lo se to F , and which ac tua lly co in c id es with F (nam e ly , with f0and f j ) on IК I = (X x{0})u (Xx{l}). It follows th a t f 0 and fj a r e hom otopica llyequ iva len t in O(oo).

P ro o f of p ro p o s i t io n 3. The m ap j induces a g roup h o m o m o rp h ismj¡ : n t (0(oo))----»• rij (0(H; Cp )) fo r a l l i e { l , 2 , . . .} and a s e t h o m o m o rp h ismbetween the n 0's . T hese h o m o m o rp h ism s a r e s u r je c t iv e by l e m m a 2 and in je c t iv e by le m m a 3. Hence, a l l the m a p s j ¡ ' s a r e i s o m o r p h i s m s , which i s e x p r e s s e d by say ing th a t j i s a "w eak homotopy eq u iva lence" . A s ta n d a rd te c h n ic a l t r i c k allows us to conclude: The inductive l im i t O(oo) has the homotopy type of an ANR (Hansen [3], c o r o l l a r y 6. 4) and the B anach m a n i fold 0 (H ; C p) i s an A.NR. Hence W h iteh e ad 's le m m a app l ies ( se e , e. g.P a la i s [11], sec t io n 6 .6 ) and j i s a homotopy equ iva lence .

4. SPIN GROUPS AND CAR

As a c o r o l l a r y to p ro p o s i t io n 3, the group 0 (H ; Cp ) has two connected com ponen ts (a lso given by the s ign of the d e te rm in a n t in the c a se p = 1); and i t s connec ted com ponent 0 +(H; Cp ) h a s Z2 as fundam en ta l group. It follows f ro m g e n e ra l p r in c ip le s th a t the u n iv e r s a l c o v e r in g Spin(H; C p ) of 0 +(H; Cp) i s again a B a n ac h -L ie group , with the s a m e L ie a lg e b ra as th a t of 0 (H ; C p). To c o n s t ru c t th is tw o-fo ld c o v e r in g exp l ic i t ly (at l e a s t ,

76 DE LA HARPE

in the e a s i e s t c a s e , i. e. p = 1), i t is p o ss ib le to p ro c e e d as follows: The

x j | x |2 can be m ade a

r e a l involutive n o rm e d a lg e b ra , with the n o rm of X e C l ( H ) being defined as | x | „ = sup {X€ IR IXX* - X2 is not in v e r t ib le} and with the involutionX ----►X* of C1(H) being the (unique) a n t i - a u to m o rp h is m of C1(H) whoser e s t r i c t i o n to H is the iden tity (H i s canon ica lly iden tif ied with a subspace of C1(H), which i s then a s y s te m of g e n e r a to r s of th is a lgeb ra ) . The c o m p le tion C l^ H ) of C1(H) with r e s p e c t to | |„ i s a r e a l C * -a lg e b ra . By the u n iv e r s a l p ro p e r ty of C lifford a lg e b r a s , any o r thogona l o p e r a to r U c O (H ) can be ex tended to a * - a u to m o rp h ism of C li(H ) which we w ill denote by C1(U).

C o n s id e r now a " ' ' -au tom orphism of Cli(H ) which m aps H onto i t s e l f (one of the s o - c a l le d "Bogoliubov a u to m o rp h ism s" ) , so tha t i t can be w ri t ten a s C1(U) fo r som e UGO(H). Then C1(U) is sa id to be in n e r if th e re ex is ts an in v e r t ib le e le m e n t u e C l i ( H ) such tha t C1(U)(X) =иХи_1 fo r a l l X e C l i ( H ) ; an in n e r a u to m o rp h ism C1(U) is sa id to be even i f u can be chosen in Cl+i(H) [w here C l^ H ) =C1Î(H) © Clj(H ) i s the canon ica l Z 2 -g ra d u a t io n of the C liffo rd a lg e b ra C li(H )].

A b as ic r e s u l t about th e se in n e r au to m o rp h ism s is due to Shale and S t in esp r in g [14]:

P ro p o s i t io n 4. If U e O ( H ) , then C1(U) is in n e r and even if and only if U e O +( H ; C i ) .

One of the i n t e r e s t s of p ro p o s i t io n 4 is i t s in te r p re ta t io n in t e r m s of the canon ica l an t icom m uta t ion r e la t io n s (CAR) of quantum ph y s ic s (see a lso Shale and S t in esp r in g [13] and Slawny [15]). As a consequence of th is , it is p o ss ib le to p rove:

P ro p o s i t io n 5. L e t Spin(H; C j ) be the group of th o se in v e r t ib le e lem e n ts u £ Cl^fH) which a r e u n i ta ry [nam ely uu* =u*u = 1], even [nam ely u E C l^ H ) ] and such th a t uHu"1 = H. Then Spin(H; C i ) can be endowed with the s t r u c tu r e of a B a n a c h -L ie group such tha t the m ap

Spin(H; C i ) ----------*- 0 +(H; C i)

H ----►H \X I— »■ uxu"1/

is the u n iv e r s a l c o v e r in g of 0 +(H; C i ).That the ra n g e of the m ap py i s p r e c i s e ly 0 +(H; Ci ) i s given by

p ro p o s i t io n 4. The d e s c r ip t io n of the B a n a c h -L ie group s t r u c t u r e on Spin(H; C i ) and the p ro o f of p ro p o s i t io n 5 a r e given in Ref. [5]. The exp l ic i tc o n s t ru c t io n s of the co v e r in g s Spin (H; Cp ) ---- »-0+(H; Cp ) (when p > l ) w illhopefully be w orked out in the n e a r fu tu re .

R E F E R E N C E S

[1] DIEU DONNÉ, J. t Fondements de l ’analyse moderne, Gauthier-V illars (1963).[2] DIEUDONNE, J . , Eléments d ’analyse, 2. G authier-V illars (1968).[3] HANSEN, V. L ., Some theorems on direct lim its of expanding sequences of manifolds, Math. Scand. 29

(1971) 5.

IAEA-SM R-1 1 /1 6 77

[4] DE LA HARPE, P . , Classical Banach-Lie algebras and Banach-Lie groups of Operators in Hilbert space, Springer Lecture Notes in M ath. 285 (1972).

[5] DE LA HARPE, P . , The Clifford algebra and the spinor group of a Hilbert space (to be published in Compositio M athem atica).

[6] DE LA HARPE, P . , On infinite-dim ensional Lie groups acting on finite-dim ensional manifolds (these Proceedings).

[7] KUIPER, N ., The homotopy type of the unitary group of Hilbert space, Topology 3 (1965) 19.[8] LANG, S ., Introduction aux variétés différentiables, Dunod(1967).[9 ] LAZARD, М ., Groupes différentiables. "Neuvième lepon" of a course to be published in book form.

[10] MILNOR, J . , Morse Theory, Princeton University Press (1963).[11] PALAIS, R. S ., Homotopy theory of infinite-dim ensional manifolds, Topology 5 (1966) 1.[12] SCHATTEN, R ., Norm Ideals of Com pletely Continuous Operators, Springer (1960).[13] SHALE, D . , STENESPRING, W .F ., States of the Clifford algebra, Ann. Math. 80 (1964) 365.[14] SHALE, D ., STINESPRING, W. F . , Spinor representations of infinite orthogonal groups, J. M ath. Mech. 14

(1965) 315.[15] SLAWNY, J . , Representations of canonical anticom m utation relations and im plem entability of

canonical transformations, Thesis, The Weizmann Institute of Science, Rehovot (1969); see also Commun. M ath. Phys. 22 (1971) 104 (same title).

IAEA-SM R-11 /1 7

THEORY O F RESIDUES IN S E V E R A L VARIABLES

P. DOLBEAULT D epartm ent of M athem atics,University of Poitiers,Poitiers, France

Abstract

THEORY OF RESIDUES IN SEVERAL VARIABLES.The standard notion of a residue a t a pole of a meromorphic 1-form on a surface extends to arbitrary forms

on analy tic manifolds, and can be expressed on more general spaces in (co)homological term s. The paper recalls the definition of the residue homomorphisms, reviews various contributions due to Leray, Norguet, the author and many others and, finally suggests several topics and problems for further studies.

0. INTRODUCTION

0.1 L e t X b e a R iem ann ian s u r f a c e and le t u b e a m e ro m o rp h ic d i f fe re n t ia l fo rm of d e g re e 1 on X; in the neighbourhood of a point w h ere z is a lo c a l c o - o r d in a te , we have w = f (z )dz w h e re f is a m e ro m o rp h ic function. L e t Y = {aj}jeI be the se t of po les of u; fo r e v e ry j G I, le t Yj be a p o s i t ive ly o r ie n te d c i r c l e w ith c e n t r e a¡ such tha t the c lo sed d isk of which is the bo u n d ary does not m e e t Y in any point d if fe ren t f ro m a¡; le t R e sa.(u) be the C auchy r e s id u e of и at a -, then fo r any f in ite su b se t J of I and fo r1 any fam ily (n j) jeJ of in te g e r s , o r r e a l o r com plex n u m b e rs , we know the fo rm u la of r e s id u e s

is a n e c e s s a r y and su ff ic ien t condition fo r a g iven s e t of n u m b e rs (Resa. )jeI to be the r e s id u e s of a m e ro m o rp h ic fo rm with po le s at the points aj.

P o in c a r e was the f i r s t to give a conven ien t g e n e ra l iz a t io n of the notion of r e s id u e s fo r c lo sed m e ro m o rp h ic d i f fe re n t ia l f o rm s in s e v e r a l com plex v a r ia b le s (1887) [27].

(1)

0.2. M o re o v e r , if X is co m p ac t the condition

(2)

79

80 DOLBEAULT

S ta r t in g f ro m P o in c a r é 's w ork , L e r a y (1959) [18] , then N orgue t (1959) [22,23] g e n e ra l iz e d the s i tua tion of s u b - s e c t io n 0. 1. to the c a s e of a com plex ana ly t ic m anifo ld X of any fin ite d im ens ion , Y being a com plex sub -m an ifo ld of X of cod im ens ion 1; u a c lo sed d i f fe re n t ia l fo rm of d e g re e p on X, of c l a s s С outside Y and having s in g u la r i t i e s of the following type on Y: e v e ry point x 6 Y has a neighbourhood U in X o v er which a com plex loca l c o -o rd in a te function s is defined such tha t YH U= {yG U; s(y) = 0} ; then и I U is equa l to a / s k w h ere a is C " on U and к 6 N. M ore g e n e ra l ly , a s e m i- m e ro m o rp h ic d i f fe re n t ia l fo rm on X has a lo c a l e x p r e s s io n a / f w here a is a C” d i f fe re n t ia l fo rm and w here f is a ho lom orph ic function.

(a) The topo log ica l s i tu a t io n s tud ied by L e r a y and g e n e ra l iz e d by N orgue t is the following:

L et X be a topo log ica l sp a ce and Y be a c lo sed su b se t of X, we have the following exac t cohom ology sequence with com plex coeff ic ien ts and co m pac t su p p o r ts

■ • • - H^(X) - HcP(Y) - HcP+1( X \ Y ) - HP+1(X) - • • • (3)

If Y and X \ Y a r e o r ie n ta b le topo log ica l m an ifo lds of d im ens ion m and n r e s p e c t iv e ly , the duality i so m o rp h is m of P o in c a ré de f ines , f ro m 6V, the h o m o m o rp h ism

6 : H° ( Y ) - H ° n(X\Y) m-p ' ' n-p-1 '

0. 3. T h eo ry of L e ra y -N o rg u e t

Hence, f ro m the u n iv e r s a l coeff ic ien t th e o re m , the r e s id u e h o m o m o rp h ism

r = l6 : H n"p’1{X\Y) Hm"p (Y)

and the fo rm u la of r e s id u e s

■(ôh, c)> = <h, r c ) (4)

in which h G H ^ . p(Y) and c G H n"p"1(X \Y ).C le a r ly , fo rm u la (4) g e n e r a l iz e s fo rm u la (1) w here с is the cohomology

c l a s s defined by ш | X \ Y and h is the homology c la s s

I Vijej

(b) Suppose now tha t X is a com plex ana ly tic m anifo ld . Given a s e m i- m e ro m o r p h ic fo rm w on X, C" excep t on a subm anifo ld Y of cod im ens ion 1,

IAEA-SM R-11/ П 81

such tha t , lo c a l ly , и = a/s w h e re s = 0 is a m in im a l lo c a l equation of Y, we sa y tha t w a d m its Y a s a p o la r se t with m u l t ip l ic i ty 1. Then , locally , и = ( d s / s ) A(//+ d0 w here Ф and в a r e С ” ; Ф | Y has a g loba l defin ition and is ca l le d the r e s id u e fo rm of и and i ts cohom ology c l a s s in Y is the im a g e , by r , of the cohom ology c l a s s defined by u | x \ Y on X \Y .

So, in th is p a r t i c u l a r c a s e , we have an in te r p re ta t io n of the r e s id u e h o m o m o rp h ism in t e r m s of d i f fe re n t ia l fo rm s .

(c) T h e o re m of L e ra y ([18] , th e o re m 1, p. 88): L et X be a com plex ana ly t ic m anifo ld and Y be a subm anifo ld of X of cod im ens ion 1, then e v e ry c l a s s of cohom ology of X \ Y contains the r e s t r i c t i o n , to X \ Y , of a c lo sed s e m i - m e r o m o r p h ic d i f fe re n t ia l fo rm having Y a s a p o la r s e t with m u lt ip l ic i ty 1. When d im Œ = 1, th is th e o re m r e d u c e s to the s ta te m e n t in s u b - s e c t io n 0 .2 .

(d) C om posed r e s id u e s : L et Yj, . . . , Yq be com plex subm anifo lds of Xof co d im en s io n 1, in g e n e r a l p o s i t io n and le t y = YjH . . . П Yq , then by c o m pos i t ion of the r e s id u e h o m o rp h ism s

HP(X\YjU . . . UYq) - H5' 1 (Yj\ Y2U. . . U Yq) - HP' 2(Yi n Y2 \ Y3U . . . U Yq )

. . . - HP 4(y)

we obtain the com posed re s id u e ; a f t e r a p e rm u ta t io n of the Y^, the com posed r e s id u e is m u l t ip l ie d by the s ig n a tu re of the p e rm u ta t io n .

0 .4 . G e n e ra l iz a t io n s . F r o m 1968, g e n e ra l iz a t io n s of the defin ition (a) of the r e s id u e h o m o m o rp h ism have been u se d fo r sp a c e s m o r e g e n e r a l than co m plex m an ifo ld s ; (b) s e m i - m e r o m o r p h ic d i f fe re n t ia l fo rm s with p o la r s e t hav ing a r b i t r a r y s in g u la r i t ie s and g e n e ra l iz a t io n s of such f o rm s have been u se d to give in te r p re ta t io n s of the r e s id u e h o m o m o rp h ism ; the th e o re m of L e r a y (c) and com posed r e s id u e s (d) have been g en e ra l iz e d ; f inally , a t te m p ts to get r e s id u e fo rm u la e have been m a d e [34, 26 ] . We sh a l l r e v ie w these r e s u l t s , except those c o n c e rn in g r e s id u e fo rm u lae .

In fac t , a s m a l l p a r t of th is p r o g ra m has been r e a l iz e d be fo re L e r a y ' s w ork (1951-57) [16 ,17 , 33, 2, 5 ] . But fo r a m o r e com ple te r e a l iz a t io n , the following n ew e r techn iques have b ee n used : B o re l -M o o re homology ([4] , c h a p te r 5); lo c a l cohom ology (G ro thendieck [13] ); H iro n a k a 's lo c a l r e so lu t io n of s in g u la r i t i e s of com plex ana ly t ic sp a c e s [15]; p r o p e r t i e s of s e m i- a n a ly t ic s e t s (fcojasiewicz [19] , H e r r e r a [9] ).

T h is p a p e r does not conta in the rev ie w of r e c e n t work on the sub jec t by N orgue t ( re s id u e s and q -convex ity [25] ), by G ordon on a g e o m e t r i c a l th e o ry of r e s id u e s [12] , by G riff i ths on r e s id u e s on a lg e b ra ic m an ifo lds and by s p e c ia l i s t s in a lg e b r a ic g e o m e t ry (G ro thendieck , H a r t s h o rn e , . . . ). Let u s a lso r e m a r k tha t r e s id u e s can be s tud ied on in f in i te -d im e n s io n a l s p a c e s [1].

1. DEFINITION OF THE RESIDUE HOMOMORPHISM

Homology and cohom ology g roups a r e supposed to be with coeff ic ien tsin(D.

82 DOLBEAULT

L et X be a loca l ly co m pac t , p a r a c o m p a c t sp a ce of f in ite d im e n s io n andY be a c lo sed su b se t of X. In the B o r e l -M o o r e homology (homology of lo c a l ly co m p ac t sp a c e s ) [4] , we have the following exac t homology sequence:

• • • ~ H q+l(X) ~ H q+1 ( X 4 Y ) 6* H q(Y) ^ H q(X) ^ ‘ ‘ ^

It is t r a n s p o s e d f ro m the exac t cohom ology sequence (3).When X \ Y and Y a r e o r ie n ta b le topo log ica l m an ifo ld s of d im ens ion n

and m , r e s p e c t iv e ly , we have the following d ia g ra m :

P x * P +iHc (Y) A*. Hc (X \Y )

H m' p(Y) J - X _ Hp(Y)

r I (*) x (Q )n-p-l n 6 = l (5

H (X \Y ) x \ Y * H (X \Y )•*--------------- p+i

1. 1. H om ological r e s id u e [8a]

w h ere p Y and p x \ Y a r e ^ e duality i s o m o rp h is m s of P o in c a ré , w here (*) is an t ico m m u ta t iv e and w h e re (Q) g ives a r e s id u e fo rm u la ; 6* w ill be ca lled hom olog ica l r e s id u e h o m o m o rp h ism and g ives a g e n e ra l iz a t io n of r .

1 .2 . C ohom olog ica l r e s id u e

F o r any topo log ica l sp a ce X and any c lo se d s u b se t Y of X, we have the exac t sequence of lo c a l cohom ology

------ - H P(X) - HP(X \Y ) - H ^ X ) - HP+1(X) - • • • (6 )

w here Ну (X) is the (p + l)- th g roup of cohom ology of X with su p p o r t in Y [13] . When X \ Y and Y a r e topo log ica l m an ifo ld s of, r e s p e c t iv e ly , d im e n s io n s n and m , the following d ia g ra m is com m uta tive :

HP(X \Y ) — ----- HP+1(X)

the h o m o m o rp h ism p is a g e n e ra l iz a t io n of r and w ill be ca l le d the cohom olo- g ic a l r e s id u e h o m o m o rp h ism .

1. 3. R e la tions be tw een the exac t s e q u en c es (5) and (6) [10]

IAEA -SM R -11/17 83

L et X be a re d u c e d com plex ana ly tic sp a ce of com plex d im e n s io n n and Y be any c lo sed su b sp a ce of X, then we have the following co m m u ta t iv e d ia g ra m :

p P П p+1(C) • • • - H (X, (С) - H (X\Y,(C) ^ H y (X, <C) - • • •

m ( x ) i n ( u ) j. n(Y) (7)

<D > • • - H 2„-p(X « Œ) - H2n.p(X\Y,<E) - H2n.p. l{Y,<C)

w here the v e r t i c a l h o m o m o rp h ism s a r e g iven by cap p ro d u c t with the fu n dam en ta l homology c l a s s of X ([10] , s u b - s e c t io n 1. 7). When X, and h ence X \ Y , is a m an ifo ld , n(X) and O(U) a r e in v e r s e s to P o i n c a r é 's duality i s o m o r p h i s m s , then n(Y) is an i s o m o rp h i s m , too [10].

2. IN T ER PR ETA TIO N OF THE MORPHISMS OF (7) WHEN codim,,. Y= 1

X deno tes a com plex ana ly tic ( reduced) sp a ce and Y a c lo sed su b sp a ce of X of p u re cod im ens ion 1. Note tha t h o lom orph ic functions , C” (or sm ooth) d i f fe re n t ia l f o rm s and c u r r e n t s can be defined on X (see Refs [3 ,10] ).

2 .1 . C a se n= 1; p r in c ip a l va lues ; r e s id u e c u r r e n t

F i r s t c o n s id e r the c a se w h ere X is a c o -o rd in a te dom ain U of a R iem ann ian s u r fa c e and le t ш be a m e ro m o rp h ic fo rm of d e g re e 1 on U having only one pole P in U. C hoose the c o -o rd in a te z on U such tha t z(P) = 0.

C o n s id e r the c u r r e n t и on U \ { P } defined in the following way: for e v e ry ф е й ( и \ { Р } ), le t

и [ ф ] = j ' и л . ф

и

T hen

V p (u ) [^ ] = l im / ъ>Лф € ~ * 0 J

| z | a e

fo r e v e ry фЕ<3(U) is a c u r r e n t on U, whose r e s t r i c t i o n to U \ { P } is u; th is c u r r e n t is independent of the choice of the c o -o rd in a te z and of the r e p r e s e n ta t ion of the m e ro m o rp h ic fo rm u; it is the s o - c a l l e d "C auchy p r in c ip a l value" of Ш. M o re o v e r ,

d V p (u ) = d" Vp(u) = 27ri7Ôp + d 'B

84 DOLBEAULT

w h e re 7 i s the Cauchy r e s id u e of 10 a t P , 6p the D ira c m e a s u r e a t P and В a c u r r e n t whose su p p o r t is in {P} ; if P is a s im p le pole of 10, we can take B = 0. The c u r r e n t d V p (u ) w ill be ca lled the r e s id u e c u r r e n t of u.

We sh a l l g e n e r a l iz e the c o n s t ru c t io n of the r e s id u e c u r r e n t to the c a se of s e m i - m e r o m o r p h ic d i f fe re n t ia l f o rm s on a com plex ana ly tic sp a c e X.

2 .2 . P r in c ip a l va lues

2 . 2 . 1 . An ana ly tic su b se t Y of a com plex ana ly t ic m anifo ld X, of com plex cod im ens ion 1, has n o r m a l c r o s s in g s if, fo r e v e ry point x e Y , th e re e x i s t s a c h a r t (z1 ( . . . , z ) on a ne ighbourhood U of x such that

A s e m i - m e r o m o r p h ic d i f fe re n t ia l fo rm defined on a ne ighbourhood U of x G X is sa id to be e l e m e n ta ry if it has a p o la r se t of the above fo rm U П Y.

2. 2. 2. On a com plex m anifo ld X, a d i f fe re n t ia l o p e r a to r D on the sp a c e of c u r r e n t s is s a id to be se m i- h o lom orph ic if, fo r ev e ry x € X ,th e r e e x i s ts a c h a r t (U,Zj, . . . , z n) a t x such tha t , on U,

w h ere the a}s a r e C°° functions on U [33] .D o p e r a te s a l so on the sp a c e ÿ(X) of the s e m i - m e r o m o r p h ic f o rm s on X.

M o re o v e r , le t A be the r in g of the s e m i-h o lo m o rp h ic d i f fe re n t ia l o p e r a to r s , then 5^(X) and S>'(X) a r e Д -m o d u le s [33].

2. 2. 3. L e t (U, z1, . . . , z n) be a c h a r t of a com plex ana ly tic m anifo ld X and Y be an ana ly tic su b se t of U whose equation is zx. . . zq = 0 (qâ n). Let

be the sp a c e of s e m i - m e r o m o r p h ic f o rm s on U of which Y is a p o la r se t and which a r e w r i t te n ar/zP w h e re a is C” on U and p= (p . . . , pn )G N n with

2. 2 .4 . In the above no ta t ions , th e r e e x i s t s a unique o p e r a to r T: </Y -» 3>'(V) such that:

(1) If wej/Y has lo c a l in te g rab le co e ff ic ie n ts , then T (u )= 10, the c u r r e n t defined by и ,

(2) T is Д - l in e a r .

M o re o v e r , T is equa l to the Cauchy p r in c ip a l value (see s u b - s e c t io n 2.1.) with r e s p e c t to ea ch of the c o -o rd in a te functions [ 6 , 7 , 8 ] .

2. 2. 5. With the no ta tions of 2. 2. 3. , le t g = zb gQ be any function w here b j . . . b4 =/= 0 and w here g0 is ho lom orph ic and without z e r o s on U. T hen , fo r e v e ry cpe^> (U), the fo rm u la

TJOY = {xG U ; z 1 ( x ) . . . z q(x) = 0; q S n }

Pq+k = O f o r k S l .

IAEA -SM R -11/17 85

Vp(u)[m] = l im à~*0 I UA ф (8 )

def ines the c u r r e n t T of 2. 2. 4.T h is def in i t ion of Vp(u) is due to H e r r e r a - L i e b e r m a n [10] .One e s ta b l i s h e s the e x is te n ce of Vp(u) fo r a p a r t i c u l a r r e p r e s e n ta t io n

of to and one v e r i f i e s tha t Vp(u) s a t i s f ie s the conditions (1) and (2) of2. 2 .4 . So, we obta in the independence of V p (u ) with r e s p e c t to the r e p r e sen ta t io n of ш and with r e s p e c t to g; m o r e o v e r , Vp(u) s a t i s f ie s conditions (1 ) and (2) of 2. 2. 4. (The idea of th is p ro o f is due to Robin [7, 8] ).

2. 2. 6. [10, 6 ,7 ] ) . L e t X be a re d u c e d com plex ana ly tic sp a ce ofco m p lex d im e n s io n n and le t Y be a com plex ana ly t ic su b se t of X of cod im e n s io n 1, con tain ing the s in g u la r locus of X, g lobally defined by an equa tion g= 0 w h ere g is h o lom orph ic o v e r X; le t f be a ho lom orph ic function o v e r X such tha t f = 0 im p l ie s g= 0; one c o n s id e r s the s e m i - a n a ly t i c se t

Let I [ X ( > 6 ) ] be the in te g ra t io n c u r r e n t o v e r X defined by the s e m i- ana ly t ic cha in [X (>ó), e (> ó ) ] w here e (> 6 ) is the fu n dam en ta l c l a s s of ( X ( > 6)).

F o r a s e m i - m e r o m o r p h ic fo rm и = a/ f (where a is C “ o v er X), we se t

fo r ev e ry ipG®(X).Let ir:X'~>X be a m o r p h i s m of H ironaka such tha t X ' be a m anifo ld and

Y 1 = 7Г ■! Y an ana ly t ic su b se t with n o r m a l c r o s s i n g s (then ir is p r o p e r and 7ГI X 1 \ Y 1 is an ana ly tic i so m o rp h is m : X '\Y '- > -X \Y ) [15]. Such a m o r p h is m e x i s ts loca lly . We have:

w h ere (X '(>6 )) = { x 'E X 1; | 7r*g(x')|>6} . F o r ir f ixed, Eq. (10) is independentof the r e p r e s e n ta t io n of и and of g f ro m 2. 2. 5. ; so , Eq. (9) m a k e s se n se and, f ro m i ts e x p r e s s io n , is independent of ir.

U sing a p a r t i t io n of un ity , Vp(u) is defined fo r e v e ry s e m i - m e r o m o r p h ic fo rm и o v er a re d u c e d com plex ana ly tic sp a ce X.

L e t ^ ( X ) be the v e c to r sp a c e of s e m i - m e r o m o r p h ic d i f fe re n t ia l f o rm s o v e r X, then V p :y (X ) -*• <£>'(X) is (D-linear.

2. 2. 7. [8]. T he r in g Д of s e m i- h o lo m o r p h ic d i f fe re n t ia l o p e r a to r scan be defined on X and it can be shown, a t l e a s t when X is a m an ifo ld , tha t Vp is a Д - l i n e a r ap p l ica t ion of Д -m o d u le s .

(X (>6 )) = { x e X ; |g (x ) | >6}

Vp(u)[cp] = l im I [X (> ó )] ( и л ф ) 6 -о

(9)

Ур(ш)[ф] = l im l [ X ' ( > ó ) ] [7г*(иЛср)] = Vp (7г*и) [ тг'ср] (10)Ó-* О

86 DOLBEAULT

2 .3 . Residue current [10, 6, 7]

We c o n s id e r the following r e s id u e o p e ra to r

R es = d Vp - Vp d ( I D

In the no ta tions of 2. 2. 6. , it h as the following lo c a l e x p re s s io n

R es (ш) [ф\ = l i m l [ W ( = ó ) ] (шЛ.<//)6^0

fo r e v e ry фЕ0(Х) and w h ere I[W (=6)] is the in te g ra t io n c u r r e n t on the s e m i- ana ly tic s e t l x £ X ; |g (x ) | = 5} with convenien t o r ien ta t ion .

2 .4 . In te r p r e ta t io n of the d ia g ra m (7) [10]

2 .4 . 1. Denote by the sh e a f com plex of C°° d if fe ren t ia l fo rm s on X, by<á^' (# Y) the sh e a f com plex of s e m i - m e r o m o r p h ic d if fe re n t ia l fo rm s w ith p o la r s e t in Y a n d b y sQ ^ the quotient sh e a f such tha t the following se quence is exact:

w h ere i deno tes the inc lusion .

2. 4. 2. L e t be the sh e a f com plex of c u r r e n t s on X a n d ^ y » » thesu b sh e a f of c u r r e n t s with su p p o r ts in Y; the following s h o r t exac t sequencedef ines g>'

• X /Y ”

o - ® : r - ~ K - K , r - - 0

P r in c ip a l v a lu es and Res define sheaf h o m o m o rp h ism s

° ^ x 1 sQ X - 0

V n - ¿ fp ( * Y ) -» 'v p - M 2n-p, X

R es: ^ p(* Y )-^> ,2 П -Р -1 , Y * °

In fac t , R es (u) depends only on the c l a s s of и m odulo , hence a h o m o m o rp h ism

Define V p ' = j • Vp w here j -*■ &'x /y °° ; f ro m Eq. (11), we get d R e s = -d V pd = - R e s d, hence a skew com plex h o m o m o rp h ism

R es: sQ x -*®.'y «

IAEA-SM R-11/ И 87

T hen we get the following d ia g ra m of cohomology

• • • - HP(X, Sx)^ HP(X, <^(*Y))-* H P(X, sQ'x ) -» ••• (A)

(*) V*Vi V p ' i Res (*)

4 » ) ^ (B)

w h e re V is defined by in te g ra t io n o v e r X; V p1 and R es a r e induced by V p 1 and R es re sp e c t iv e ly ; the s q u a re s of th is d ia g ra m a r e com m u ta t iv e , excep t (#) which a r e an t ico m m u ta t iv e .

2. 4. 3. F in a l ly , we have the d ia g ra m

w h e r e the a r r o w s (A) -*■ (C) and (B) ->■ (D) com e f ro m m o r p h i s m s of de R h a m 's th e o re m in cohom ology [3] and hom ology [10]. The s q u a r e s a r e co m m u ta t iv e except one o v e r th r e e in (A) -» (B) and in (A) ->■ (C) which a r e an t ico m m u ta t iv e . M o re o v e r (B) -* (D) is su r je c t iv e .

T h is d ia g ra m g ives an in te r p re ta t io n of the r e s id u e h o m o m o rp h ism for hom ology o r cohom ology c l a s s e s in X \ Y which can be defined by r e s t r i c t i o n of s e m i - m e r o m o r p h ic d i f fe re n t ia l f o rm s . M o re o v e r , H e r r e r a - L i b e r m a n [10] have a s i m i l a r r e s u l t in which the exac t sequence (A) is r e p la c e d by the co r r e sp o n d in g exac t hypercohom ology sequence fo r m e ro m o rp h ic d i f fe re n t ia l fo rm s with p o la r s e t on Y.

2 . 4 . 4 . When X and Y a r e m a n ifo ld s , then the m o r p h i s m s of d ia g ra m (12) a r e a l l i s o m o rp h i s m s and, if r e s ( u ) deno tes L e r a y ' s r e s id u e c l a s s of the s e m i - m e r o m o r p h ic d if fe r e n t i a l f o rm w, then Res (u) = 2 7ri l [ Y ] n r e s (w).When X \ Y is a m a n ifold, V p 1 s p l i t s canon ica lly . When X is a m an ifo ld , V is an i s o m o rp h is m , Res and Vp' sp l i t canon ica lly .

2. 5. I n te r p r e ta t io n of 6^ fo r s e m i - m e r o m o r p h ic d i f fe re n t ia l f o rm s with p o la r s e t in Y is g iven by m e a n s of homology of open ne ighbourhoods of Y, with app l ica tion to the r e s id u e of a C ous in da ta in ( [5a], 5, 6])

(1 2 )

88 DOLBEAULT

3. EXACT SEQUENCES (B) AND (D) WHEN X IS A MANIFOLD [32] ; RESIDUE CURRENT IN ANY DIMENSION [31]

3. 1. Let X be a r e a l ana ly t ic m an ifo ld and Y a c lo sed ana ly tic su b se t , then the following exac t seq u en ces a r e i som orph ic :

Hqr(X,Æ>'x ) - Hqr (X, Æ>!x /T - ) - H Г (X, S>'. y«°)xj.

H (X, <E )q

X/Y ;

Hq( X \ Y, CD) ■

Y 1

H ^ Y . Œ ) -

(В)

(D)

w h ere the т 's a r e the h o m o m o rp h ism s of the hom olog ica l de R h a m 's th e o re m .

3. 1. 1. Sketch of the proof. t x is an iso m o rp h is m s in c e X is a m anifold .(a) It su f f ice s to p ro v e the th e o re m locally ;(b) the th e o re m is t r u e when Y i s a c lo sed subm anifo ld of X (use a

homotopy fo rm u la fo r c u r r e n ts ) ;(c) the th e o re m is t r u e when Y is a h y p e r s u r fa c e with n o r m a l c r o s s in g s

(use the exac t sequence of M a y e r -V ie to r i s val id fo r c lo sed s e t s in r e g u la r pos i t ion (Lo jas iew icz [21], C h a p te r 7, p rop . 1 .4));

(d) to obtain the g e n e r a l c a s e , u se a m o r p h is m of H ironaka [15] 7r:X->-X, w here тг-1 Y = Y has n o r m a l c r o s s in g s . C o n s id e r the following d ia g ra m :

НаГ(Х,^> ' .~ „„) а НаГ ( Х , ^ ' . »)4 X / Y --- ► 4 X / Y

T ~ i T IX \ Y X \ Y

Hq(X \Ÿ , Œ) Hq(X \Y , (C)

w h ere a and (3 a r e induced by n . F r o m (с), t ~ is an i s o m o rp h is m , hence a l s o t„ », ; )3 is an iso m o rp h is m s ince 7 r |x \Ÿ is an i so m o rp h is m ; to show tha t t x is an i s o m o rp h is m (which w ill e s ta b l i s h the th e o re m ) and s ince F (X , ¿Ь''х/y " ) is m ade of the c u r r e n t s on X \ Y which have an ex tens ion over X, it is enough to p ro v e that

7Г*: Г ( Х , ^ > ' J - r ( X , ^ ' )' x -x

is s u r j e c t iv e . It su f f ice s to p rove tha t 7r*: ,ÿn(X) -► ^ n(X) is in jec t ive with c lo sed im age; th is can be done by u s in g the d iv is ion of a d i s t r ib u t io n by a r e a l ana ly tic function (Lo jas iew icz [21], C hap te r 7).

3. 2. R esidue c u r r e n t on a r e a l ana ly tic m an ifo ld [31]

Let X be a r e a l ana ly t ic m an ifo ld of d im e n s io n m , d en u m erab le at infinity and le t Y be a c lo sed s e m i- a n a ly t ic su b se t of X of d im ens ion I < m . F o r s im p l ic i ty , we sh a l l a s s u m e tha t X is o r ien tab le .

IAEA-SM R-11 /17 89

3. 2. 1. L e t ^ . ' x be the sh e a f com plex of the s e m i - a n a ly t i c cha ins on X. We define 'ÿ'.'y” and ^ ' . x / y” as f ° r c u r r e n t s . In teg ra t io n of sm oo th d i f fe re n t i a l f o rm s on s e m i- a n a ly t ic cha ins def ines a m o r p h is m of sh eav es

I : « V ^ ' x

A c u r r e n t T of d im ens ion q on X \ Y is sa.id to be loca lly bounded on X if, fo r e v e ry Yël^, (X, ), the d is t r ib u t io n T A j ''~7 is a bounded m e a s u r e onX \ Y (j deno tes the inc lusion : X \ Y -*X). It is a n e c e s s a r y and su ff ic ien t condition fo r the ex is te n c e of a 0 - continuous ex tens ion T of T on X (this notion is due to Lelong [20] ).

L e t Bq(X, Y) be the sp a ce of c u r r e n t s T on X \ Y of d im ens ion q such tha t T and dT be loca lly bounded on X; we define the m o r p h i s m r e s id u e - c u r r e n t by r e s T = dT - dT; we have: r e s dT = -d r e s T.

Now, we can w r i te the following com m uta t ive d ia g ra m (where r e s is induced by r e s ) :

H m' q Г (X, <rx)

H q(X,<C)

The i s o m o rp h is m is de R h a m 's fo r m an ifo ld s and irY ' Iy is an iso m o rp h is m .

3. 2. 2. A d i f fe re n t ia l f o rm w ill be s a id loca l ly in te g rab le if i ts co e f f ic ien ts a r e loca lly in te g rab le .

3. 2. 3. E v e r y o e Hq( X \Y , CD) con ta ins a (m -q ) - fo r m which is sm oo th and c lo sed on X \ Y and hav ing the following p r o p e r t i e s :

(i) th e re e x i s ts a loca l ly in te g ra b le fo rm cp on X whose r e s t r i c t i o n to X \ Y is cp;

(ii) the r e s i d u e - c u r r e n t of cp is a s e m i - a n a ly t i c cha in on Y.The p ro o f u s e s d ia g ra m chas ing and the following lem m a:

3 . 2 .4 . On the com plex £>"{X) of c u r r e n t s on X, th e re e x is t continuous l in e a r e n d o m o rp h ism s A and R with d e g re e -1 and 0 r e s p e c t iv e ly such that:

(i) fo r e v e ry c u r r e n t T, we have T = dAT + AdT + RT;(ii) RT is sm ooth ; s in g supp AT С s in g supp T;

(iii) if T is О-co n t in u o u s , then AT is a loca lly in te g rab le fo rm .

4. L E R A Y 's THEOREM

In th is p a r a g ra p h , X is a com plex ana ly t ic m anifo ld of d im e n s io n n and Y is a c lo sed ana ly t ic su b se t of X of p u re cod im ens ion к г 1.

HqB .(X ,Y ) ___i N ^ e s I Y;

H q Г ( X , ^ ' - Г 4 Н ^ П Х , ^ - )7Г

i Yi

H ( X \ Y , Œ) 5 * H q- i ( Y , Œ) ( D )

90 DOLBEAULT

4. 1. 1. [28, 28a] Suppose tha t Y is an an a ly t ic su b se t with n o rm a l c r o s s i n g s , then the th e o re m of L e r a y is t r u e , i .e . fo r any c l a s s of cohom ology a in HP(X \Y , (E), th e re e x i s ts a p - s e m i - m e r o m o r p h i c fo rm on X whose p o la r s e t is Y with m u l t ip l ic i ty one and whose r e s t r i c t i o n to X \ Y belongs t o a .

The p ro o f u s e s the following le m m a s and a c l a s s i c a l r e s u l t of sheaf theory :

(a) Let ux be an e le m e n ta ry fo rm around x £ X ; (Zy, . . . , zn) be loca lc o - o r d in a t e s such tha t the p o la r se t of ux is contained in Z j . . . zn = 0. LetI = (in, . . . , i J e i l, ,n}; wë s e t z , = z, . . . z¡ and dz, = d z . A . . . Adz. , thenJ 1 * 1 Ak 1 1 к

u = ф + d x

w h ere ¡í is e l e m e n ta ry and w here

4. 1. C a se к = 1

V = (dz, / z jJA ^ j + >

iep([l......n})

with (//j and AC in a neighbourhood of x.(b) L et ux be a g e r m of c lo sed e le m e n ta ry fo rm ;

(i) if d e g re e ux > dim^-X, then ux is exact;(ii) if d e g re e wx S d im ŒX, then ux is cohom ologous to a g e r m of

m e ro m o rp h ic fo rm .

4. 1. 2. [28]. The above r e s u l t is va l id fo r m e ro m o rp h ic f o rm s if X is a Stein m anifo ld .

4. 1. 3. R e c a l l the following th e o re m ([14] , th e o re m 2) whose p roo f u s e s H iro n a k a 's r e so lu t io n of s in g u la r i t i e s

w h e re f :X \Y -* X is the; canon ica l in jec t ion and f2'x (*Y) is the sh e a f of m e r o m o rp h ic f o rm s on X with p o la r s e t con tained in Y.

4 . 1 . 4 . U sing th e o re m 4 . 1 . 3. , G ro thend ieck p roved a th e o re m ([14], C o ro l la ry ) w hose p a r t i c u l a r c a s e s a r e : If X is a Stein m anifo ld , o r if X is a p ro je c t iv e a lg e b r a ic m an ifo ld and Y is the su p p o r t of an am p le pos it ive d iv is o r of X, then th e r e e x i s ts a ca n o n ic a l i so m o rp h is m :

НЧГ (X, f i ' ( * Y ) ) H 4(X \Y , <0 X x4 . 1 . 5 . U sing th e o re m 4. 1. 3. and le m m a (b) of 4 . 1.1. , Robin p roved [28]

(co m p a re a l s o 2. 4. 4. ):L et Y be any ana ly tic su b se t of co d im en s io n 1 of X, then any c l a s s of

cohom ology of HP(X \Y ) conta ins the r e s t r i c t i o n to X \ Y of a c lo sed s e m i- m e ro m o rp h ic fo rm on X whose p o la r s e t is con ta ined in Y (but with no condit ions on the m u l t ip l ic i t i e s of the po les) .

4. 2. G e n e ra l c a s e ( к г 1) [29] (see Refs [23, 24, 34] )

IAEA-SM R-11 /1 7 91

4. 2. 1. K e r n e l s . L e t ly be the in te g ra t io n c u r r e n t defined by Y. We c a l l k e r n e l (or q u as i -k e rn e l . , r e sp e c t iv e ly ) a s s o c ia te d to Y ev e ry (2 k - 1 ) -d i f f e re n t i a l fo rm К loca lly in te g ra b le on X, with s in g u la r suppo r t in Y such tha t l y - d K is a sm oo th (or loca l ly in te g ra b le , r e sp e c t iv e ly ) fo rm on X.

4 .2 .2 . E x is te n c e of k e r n e ls . L e t X be a sm oo th r e a l m anifo ld which is d e n u m e ra b le at infinity and Y be a c lo se d su b se t of X. T hen , fo r ev e ry c u r r e n t T su c h that:

(i) T is 0 - continuous;(ii) d f~ ^ 0;

(iii) supp s ing T C Y , th e r e e x i s ts a d i f fe re n t ia l fo rm К loca l ly in te g ra b le on X, with s in g u la r su p p o r t contained in Y and such tha t T - dK is a sm oo th d i f fe re n t ia l fo rm on X.

4. 2. 3. A d i f fe re n t ia l fo rm в defined on an open su b se t U of X is ca lled q u a s i - s m o o th if 6 | u \ Y is sm oo th and if в and d6 a r e loca lly in te g rab le on U.

Let£f"x .Y be the sh e a f com plex of q u a s i - s m o o th d i f fe re n t ia l f o rm s on X.

4. 2. 4. The p ro o f of 4. 2. 2. is a consequence of the following two le m m a s :

(a) F o r e v e ry g e r m of c u r r e n t T of d e g re e p ï 1, 0 - continuous and d - c lo s e d , th e re e x i s ts a g e r m of ( p - l ) - d i f f e r e n t i a l fo rm K, loca lly in te g ra b le , sm o o th ou ts ide the s in g u la r su p p o r t of T and such tha t T = dK (this is a P o in c a r e lem m a ).

(b) The inc lu s ions S 'x -*-¿2 x .Y ^ x induce i s o m o rp h is m s in cohom ology

НрГ (Х ,< Г х) 2 НРГ ( Х , ^ Х;Т) 2 HPr(X,¿Z>'x )

4 .3 . G e n e ra l c a se (continuation) : q u a s i - s im p le f o rm s [29]

4. 3. 1. L e t £ P(X;Y) be the su b sp a ce of T ( X \Y , <£х) of the p - fo r m s such that

ф = (К Аф + 0)| X \ Y (13)

w h ere К is a q u a s i - k e r n e l a s s o c ia te d to Y, Ф a (p-2k +1 ) -sm o o th fo rm on X and в a q u a s i - s m o o th p - f o r m on X; th is defin ition does not depend on the cho ice of K. T h o se f o rm s a r e sa id q u a s i - s im p le on X.

4 . 3 . 2 . L et r e s : EP(X;Y) -» Г (Y, s\ 2k+1) be the m o r p h is m which, in the no ta tions of (13), is defined by

r e s cp = i"ip (14)

w h e re i is the inc lusion : Y -X ; rescp does not depend on the e x p r e s s io n of cp and is ca l le d the r e s id u e - f o r m of cp.

F o r e v e ry cpe£p(X;Y), we have dcp£ EP+1(X;Y) and r e s (dcp) = - d (rescp).

92 DOLBEAULT

4. 3. 3. ц be ing the r e s t r i c t i o n m o r p h is m , then the following sequence is exac t

w here the s q u a re s c o m m u te , except (*) which an t ico m m u te s ; the v e r t i c a l m o r p h i s m s a r e defined by in te g ra t io n o v er X, X \ Y and Y, r e sp e c t iv e ly .

4 . 3 . 4 . F r o m de R h a m 's th e o re m fo r ana ly tic sp a c e s [3 ] , we have m o r p h i s m s

4. 3. 6. A m o r e p r e c i s e th e o re m can be obta ined, but without a d ia g ra m of the fo rm (15). N am ely , К being a k e r n e l a s s o c ia te d to Y, le t ff"^(X;Y) be the su b sp a ce of £ P(X;Y) m a d e of the sm o o th p - f o r m s ф on X \Y such tha t ф = (К Л ^ + б)| X \ Y w h e re Ф and в a r e sm o o th on X. The f o rm s of (^^(Х;У) a r e sa id to be K -s im p le on X.

О - Г ( Х , £ Г х . у ) Ü r / ( X ; Y ) - S r ( Y , ^ ‘Y) - 0

T hen we have the following d ia g ra m

------► Hpr ( X , ^ ' X;Y) £ H P£ ( X ;Y )(15)

e : HP(Y,(E) - H pF (Y ,e?‘Y)

which have ca n o n ic a l r e t r a c t io n s

I : H P r ( Y , tf’y) -* HP(Y, <D) (then I - e = id)

The following d ia g ra m is com m uta t ive

(Y, <C)

(16)

w h ere t y co m es f ro m de R h a m 's th e o re m in homology. F r o m E q s (15) and (16), we get

4. 3. 5. The following two conditions a r e equivalent:

a e H2n (X \Y , Œ) contains a c lo sed fo rm ф e EP(X;Y)

6.. a = [Y] П|3 w h e re )3 G Hp"Zk+1(Y, <C) (ii)

(i)

IA EA -SM R-11/17 93

4. 3. 7. In the above no ta t ions , the following two conditions a r e equivalen t:(i) a e H 2n.p( X \ y , ( [ ) con ta ins a c lo se d p - f o r m <р€5^^(Х;У);

(ii) th e re ex is ts )3GHP 2k+1(Y,(C) such tha t = [Y]n(3.Condition (ii) is not a lways s a t is f ie d a s can be s e e n for: X = Œ2 and

Y = { ( z j , z 2 ) e ( C 2 ; Z j Z 2 = 0} .The p ro o f of the th e o re m e s s e n t i a l ly u s e s the following r e s u l t :Let ¡3 G Hp‘2k+1(Y, Œ) such tha t = 0 (i being the inc lusion : Y-*-X),

then ev e ry u 6 r ( Y , | ,pY' 2W ) belong ing to e)3 is the r e s id u e - f o r m of a c lo sed fo rm cpe<r£(X;Y).

4. 3. 8. C o ro l la ry : Let Y be a subm anifo ld of X, then ev e ry e lem e n tof H2n_p(X\Y, (E) contains a c lo sed fo rm belonging to CT£(X;Y).

If Y is of cod im ens ion 1, with lo c a l equation s = 0 and if the k e r n e l К is c o n s t ru c te d f ro m the lo c a l k e r n e ls ( 1 /2 7ri) ( d s ) / s , then 4. 3. 8. is the c l a s s i c a l th e o re m of L e ra y .

4. 3. 9. R e m a rk : In th e o re m s 4. 3. 5. a n d “4. 3. 7. , the c lo sed p - fo r m cp is co n s id e re d to be a c u r r e n t of d im e n s io n 2n-p and, in th is way, defines a c l a s s of homology.

5. COMPOSED RESIDUES

They have been defined and s tud ied p re v io u s ly in ([18, 23, 35] ).

5. 1. M a y e r -V ie to r i s m o r p h i s m [30]

5. 1. 1. Let X be a p a ra c o m p a c t r e a l ana ly tic sp a c e , Yj, Y2 be c lo sed s e m i - a n a ly t i c s u b se ts of X; le t Y = Yj U Y2 and y = Yj П Y2.

The M a y e r -V ie to r i s m o r p h i s m /j* i s defined in the exac t sequence

Hq(y,Œ) - Hq(Yj, (E) © Hq(Y2,Œ) - Hq(Y, Œ)"-*

and it is r e l a t e d to the r e s id u e h o m o m o rp h ism 6% and the com posed r e s id u e h o m o m o rp h ism 6.J.2 defined a s in L e r a y ' s p a p e r s [18,35] by the following co m m u ta t iv e d ia g ra m :

Hq(X\Y,(C)

6* ;

Hq-].(Y, Œ) —

5 .1 . 2. In te r p r e ta t io n of by m e a n s of s e m i- a n a ly t ic cha in s . L em m a. L et be the sheaf com plex of s e m i - a n a ly t i c cha ins on X, then the following se quence is exact:

0 - Г (XM'x) “ Г ( X x) © Гу2( Х ,^ . 'х ) ® rY( X , ^ . 'x ) - 0 (17)

94 DOLBEAULT

w here a= i{ - i^; P = ii+i2; i ' k and ik being defined by the inc lu s ions : y ^ Yk and Yk->Y (k = 1 ,2 ) re sp e c t iv e ly ; (this com es f ro m r(Z,çg.’z) = r z(X, eg'.x) if Z is a c lo sed s e m i- a n a ly t ic s e t of X).

5 . 1 .3 . F r o m Ref. [3] , t h e o re m 2. 8, we obtain the r e su l t :The following d ia g ra m w here v e r t i c a l m o r p h i s m s a r e i s o m o rp h is m s is cu m m u ta t iv e :

• • — Н Г ( Х , « ' х ) - H r Yi( X , ^ ' . x ) © H q r Y2( X , ^ ' . x ) - H qr Y( X ^ ' . x )

i * i (18)- Hq (y, Œ) — Hq(Y1;Œ) © Hq(Y2,Œ) - Hq(Y ,Œ )-------

5. 1 .4 . If X is a m anifo ld , the s e m i- a n a ly t ic su b se ts Yj and Y2 being in r e g u la r pos it ion , we have the analogous of (17) fo r c u r r e n t s ([21 ] , c h a p te r 4, p ro p o s i t io n 1 . 4 . ) and the i so m o rp h is m of exac t se q u en c es of 5. 1. 3. can be fa c to r i z e d th rough a m o r p h i s m f ro m the exac t sequence (18) to be c o r r e s ponding exac t sequence fo r c u r r e n t s .

5. 1. 5. R e m a rk : T h is th e o ry can be done for any f in ite n u m b e r ofc lo se d s e m i- a n a ly t ic su b se ts of X.

5 .2 . M ultip le r e s id u e s [11]

5. 2. 1. L e t X be a com plex ana ly tic sp a ce of d im e n s io n n and Y0 ,. . . , Y$ (s+1) com plex ana ly tic su b se ts of X of cod im ens ion 1; le t Y = n Y¡ (i e i = ( 0 , , . , , s ) ). L e t Д1 be the se t of the t - s im p l ic e s S ~ {iQ, . . . i t } С Д and $^(*Y S) the sh e a f of the q - s e m i - m e r o m o r p h i c d if fe ren t ia l f o rm s on X w hose p o l a r se t is contained in Ys = U (Y¡; i G S). We def ine the b ig ra d e d sh e a v e s Ct,q(0 § t S s; 0 5 q 5 n) as the sh e a v e s of t - C e c h cochains with v a lu es in the ^ X(*YS) in the following way: fo r e v e ry open se t W in X

C t,q(W)= H tr ( w , ^ Ys ))S e Д

C *’* is a double com plex , we c o n s id e r the to ta l com plex

e-£e-Ш

with

t + q = m

The sh e a f S is a su b sh e a f of С

IA EA -SM R-11/17 95

We define the sh e a f co m plex Q’ by the s h o r t exac t sequence

0 - S' - c ’ - Q ' - 0 (19)

T he exac t cohom ology sequence (A 1) a s s o c ia te d to (19) is ana logous to the se quence (A) (2 .4 . 2. ), and us ing in te g ra t io n , we can p ro v e tha t th e re e x i s t s a m o r p h i s m (A 1) -* (C) which is an i s o m o rp h is m if X is a m anifo ld (in fac t , one s q u a re o v e r th re e is an t icom m uta t ive ) .

5. 2. 2. Suppose now th a t Y i s a com ple te i n te r s e c t io n ; Д= (0 , . . . , p-1) and codim Y = p, then we can define i t e r a t io n of Vp and of R es of 2. 2 and2. 3. and p rove the e x is te n ce of the d ia g ra m analogous to the d ia g ra m (12)(2. 4. 3. ).

The m u lt ip le r e s id u e s defined in tha t way a r e r e l a te d to the com posed r e s id u e s and in the c a s e of a com ple te in te r s e c t io n Y give in te r p re ta t io n s of the r e s id u e h o m o m o rp h ism s .

The p r e s e n t p a r a g ra p h con ta ins a work by Colef to be published.As in Ref. [10] , one can u s e un iquely ho lo m o rp h ic and m e ro m o rp h ic

d i f fe re n t ia l f o rm s in s tead of C“ and s e m i - m e r o m o r p h ic fo rm s and then r e p la c e cohom ology by hypercohom ology .

The m u lt ip le r e s id u e s a r e u se fu l to c o n s t ru c t the dual iz ing com plex u sed in the p roof of the duality th e o re m fo r ana ly tic sp a c e s (see a fo r th com ing note by H e r r e r a - R a m is -R u g e t and the p a p e r of Ruget, Com plexe du a l isa n t et r é s id u s (Jo u rn é es de g é o m é tr ie ana ly tique , P o i t i e r s juin 1972). F o r the th e o re m of dual i ty , s e e the p a p e r by R a m is , Ruget and V e r d ie r , D uali té r e la t iv e en g é o m é tr ie ana ly tique com plexe , Inventiones m ath .13 (1971) 261.

6. PROBLEM S

6. 1. To g e n e ra l iz e P o ly 's r e s u l t s u s in g k e r n e ls (4. 2. , 4. 3. ) to ana ly t ic sp a c e s .

6. 2. What about the g row th of k e r n e ls in the neighbourhood of Y?

6. 3. To find a m o r e com ple te r e a l ana ly tic th e o ry of r e s id u e s .

6. 4. To study m e ro m o rp h ic and s e m i - m e r o m o r p h ic d if fe ren t ia l fo rm s with po les of m u lt ip l ic i ty one and th e i r g e n e ra l iz a t io n s in the s e n s e of Po ly (4. 3. ); in p a r t i c u l a r , to what ex ten t does the th e o re m of L e r a y r e m a in v a l id fo r Y (and X) with s in g u la r i t i e s ? (co m p a re with R ob in 's r e s u l t s (4.1)).

6. 5. To c o n s t ru c t a s e m i - m e r o m o r p h ic d i f fe re n t ia l fo rm with g iven r e s id u e c u r r e n t (p a r t ia l r e s u l t s have b ee n obtained; s e e Ref. [6] n. 4. 3, and Ref. [7] , exposé du 28. 1. 70, n. 5. 2. )

6. 6. When X is a com plex ana ly tic m an ifo ld and when Y is a p a r t i c u l a r c a s e of eq u is in g u la r ana ly t ic s u b se t of cod im ens ion 1, Vp(u) and the r e s id u e c u r r e n t of и (for a c lo se d m e ro m o rp h ic f o rm u>) have been defined in Ref. [5] , c h a p te r 4 ,D ); when Y has n o r m a l c r o s s i n g s , r e s u l t s have been g iven in Ref. [7] , 28, 1. 70, n. 4. 1. T h is study of the r e s id u e c u r r e n t has to be m ade in the g e n e r a l c a se .

96 DOLBEAULT

6.7. To find new app l ica tions of the ex is te n c e th e o re m of k e r n e ls (4.2.2). Among the p ro b le m s given in Ref. [7], 28 .1.70, n .5 .3 . , n n .2,3 ,4,5 a r e

now solved.

R E F E R E N C E S

[1] ASADA, A ., Currents and residue exact sequences, J.Fac.Sci.Shinshu Univ. 3 (1968) 85.[2] ATIYAH, M .F ., HODGE, W .V .D ., Integrals of the second kind on an algebraic variety, Ann. Math. 62

(1955) 56. —[3] BLOOM, T . , HERRERA, M ., De Rham cohomology of an analytic space, Inventiones Math. 7 (1969) 275.[4] BREDON, G .E ., Sheaf Theory, M c-Graw-Hill Series in Higher M athematics (1967).[5] DOLBEAULT, P., Formes différentielles e t cohomologie sur une variété analytique complexe, Ann. Math.

64 (1956) 83; 65 (1957) 282.[6] DOLBEAULT, P ., Résidus e t courants, C .I .M .E . Sept.1969 (Questions on algebraic varie ties); Espaces

analytiques, Sém .Bucarest (1971).[7] DOLBEAULT, P ., in Sém .P . Lelong (1969, 1970, 1971) Lecture Notes 116, 205, 275.[8] DOLBEAULT, P ., Valeurs principales sur un espace analytique, Conv.di Geometría, Milano 1971, à

paraître à l*Accademia dei Lincei.[8a] DOLBEAULT, P ., Theory of residues and homology, Is t.naz . di a lta m at.Sym posia M ath .3 (1970) 295.[9] HERRERA, М ., Integration on a sem i-analy tic set, Bull. Soc. M ath. France 94 (1966) 141.

[10] HERRERA, М ., LIEBERMAN, D ., Residues and principal values on complex spaces. M ath.Ann. 194(1971) 259.

[11] HERRERA, М ., Résidus m ultiples sur les espaces complexes, Journées complexes de M etz( février 1972).[12] GORDON, G .L ., The residue calculus in several complex variables (to be published).[13] GROTHENDIECK, A ., Local cohomology, Lecture Notes 41 (1967).[14] GROTHENDIECK, A ., On the de Rham cohomology of algebraic varieties, I.H .E .S .P ub l.M ath .N o .29

(1966) 95.[15] HIRONAKA, H ., Resolution of singularities of an algebraic variety over a field of characteristic zero,

Ann. M ath. 79 (1964) 109.[16] KODAIRA, K ., The theorem of Riemann-Roch on com pact analytic surfaces. Am. J. M ath. 73 (1951) 813.[17] KODAIRA, K ., The theorem of Riemann-Roch for adjoint systems on 3-dim ensional algebraic varieties,

Ann.M ath. 56 (1952) 298.[18] LERAY, J., Le calcul différentiel e t intégral sur une variété analytique com plexe. Problème de Cauchy 3,

Bull.Soc. Math. France _87 (1959) 81.[19] #OJASIEWICZ, S ., Triangulation of sem i-analy tic sets, Ann.scuola no rm .sup .Pisa (3) 18 (1964) 449.[20] LELONG, P ., Integration sur un ensemble analytique complexe. Bull.Soc. M ath.France 85 (1957) 239.[21] MALGRANGE, B., Ideals of differentiable functions, Tata Institute, Bombay, N o.3, Oxford Univ.

Press (1966).[22] NORGUET, F ., Sur la théorie des résidus, C .R . A c.Sci.Paris 248 (1959) 2057.[23] NORGUET, F ., Dérivées partielles e t résidus de formes différentielles sur une variété analytique complexe,

Sém .P . Lelong, 1958-59, N o .10 (24 pages).[24] NORGUET, F ., Sur la cohomologie des variétés analytiques complexes e t sur le calcul de résidus, C .R . Ac.

S c i. Paris 258 (1964) 403.[24a] NORGUET, F ., Sém .P.Lelong (1970) Lecture Notes 205.[25] NORGUET, F ., Résidus e t q-convexité. Colloque sur les fonctions de plusieurs variables complexes,

Paris (juin 1972).[26] KING, J .R ., A residue formula for complex subvarieties, preprint 1972.[27] POINCARE, H ., Sur les résidus des intégrales doubles, Acta Math. 9 (1887) 321.[28] ROBIN, G ., Formes sem i-méromorphes e t cohomologie du com plém entaire d*une hypersurface d’ une

variété analytique com plexe, C .R. A c .Sci.Paris 272 (1971) A -33-35.[28a] ROBIN, G ., Sém .P.Lelong (1970) Lecture Notes 205.[29] POLY, J .B ., Sur un théorèm e de J. Leray en théorie des résidus, -C .R .A c .Sci.Paris 274 (1972) A-171.[30] POLY, J .B ., Morphismes de M ayer-Vietoris e t résidu composé, à paraître.[31] POLY, J .B ., in Thèse en cours.[32] POLY, J .B ., Cohomologie locale e t courants. Journées de Géométrie analytique, Poitiers (juin 1972).[33] SCHWARTZ, L ., Courant associé à une forme différentielle méromorphe sur une variété analytique

com plexe, C o ll.in t.C .N .R .S ., Géométrie différentielle, Strasbourg (1953) 185.[34] SHIH WEI SHU, Une remarque sur la formule de résidus, B ull.A m er.M ath .Soc. 76; (1970) 717.[35] SORANI, G ., Sui residui delle forme d ifferenziali di una varietà analitica complessa, Rend. Mat. 21 (1962) 1.

IA EA -SM R -11/18

ON CONNECTIONS

S. DOLBEAULT Department of Mathematics,University of Poitiers,Poitiers, France

Abstract

O N CONNECTIONS.This paper covers the following topics: — Preliminaries: vector fields on a manifold, Lie groups;

connections in a principal fibre bundle: subspaces of tangent spaces, connections, parallelism, curvature and torsion, linear and affine connections, metric connections; connections and characteristic classes: Weil homomorphism, Chern classes.

T h e p u rp o se of th is p ap e r is to give an e le m e n ta ry su rv e y of the c l a s s i c a l th e o ry of c o n n e c t io n s . T h is th e o ry a r i s e s f ro m p a p e r s of C a r tan [2 , 3]; i t w as th e n developed by E h re s m a n n , [8 , 9], and it ha s been the su b je c t of m any books am ong which we sha ll only r e f e r to th o se of C h e rn [ 7] , L ic h n e ro w icz [ 13] and P h a m Mau Quan [ 16] .

T he o b je c ts un d er co n s id e ra t io n a r e d if fe ren t ia b le m an ifo lds on which we f i r s t study v e c to r f ie lds and th e ac tion of a L ie group (see L ichnerow icz [ 14]); we then c o n s t ru c t bundles o v er M, e s p e c ia l ly p r in c ip a l bundles and v e c to r bund les (for the th e o ry of f ib re b und le s , see H u se m o l le r [1 1 ] ) .Then, we sha ll give the g e o m e tr ic a s p e c t of a connection and, on having defined co v a r ia n t d if fe ren t ia t io n , we sh a l l in tro d u c e c u rv a tu re and to r s io n . We sh a l l not speak about ho lonom y g ro u p s .

A s a conclus ion , we sh a l l t r y to expose the re la t io n sh ip between d if fe ren t ia l g e o m e try and cohom ology on a m an ifo ld . To do th is , we need c h a r a c t e r i s t i c c l a s s e s . T h e se c l a s s e s w ere in tro d u c ed by P o n try a g in ,S tiefe l and W hitney. They a r e th e ob jec t of m any p a p e r s of C hern [ 6] . H i r z e b ru c h [ 10] gave the ax io m a t ic defin ition of C h e rn c l a s s e s ; we sha ll a lso r e f e r to a p ap e r by M ilnor [ 15] . The re la t io n be tw een connections and c h a r a c t e r i s t i c c l a s s e s r e s u l t s f ro m the W eil h o m o m o rp h ism which was f i r s t s tud ied by C a r ta n [4 , 5 ] , and m ay be u sed to study the ho lom orph ic s e c t io n of a H e rm i t ia n v e c to r bundle (Bott and C h e rn [ 1 ]) .

F o r the whole of th is p a p e r , we r e f e r to K obayash i and Nomizu [ 12] .T h e m a n ifo ld s , m app ings , e tc . a r e a l l supposed d if fe ren t ia b le of c l a s s

C°°, u n le s s s ta te d o th e rw is e .

1. PRELIM INARIES

1 .1 . V e c to r f ie ld s on a m anifold

a) D ef in i t io n s : A d if fe o m o rp h ism of a d if fe ren t ia b le m an ifo ld M onto i t s e l f i s ca l le d a d if fe ren t ia b le t r a n s f o r m a t io n of M, o r , s im ply , a t r a n s f o r m a t io n of M.

97

98 DOLBEAULT

A 1 - p a r a m e te r g roup of t r a n s f o r m a t io n s of M is a m app ing of IR X M into M:

R X M -----►M

(t ,x ) I-----►<pt (x)

which s a t i s f ie s the following conditions:(1) fo r e a c h t e l R , tpt : M -----► M, i s a t r a n s f o r m a t io n of M, and <p0 i s the

iden ti ty .(2) fo r a l l t , s e ® , and x e M , tpt + s (x) = tpt {tps(x))E a c h 1 - p a r a m e te r g roup of t r a n s f o r m a t io n s tpt induces a v e c to r f ie ld X as fo llow s: fo r ev e ry point x of M, X xis the v e c to r tangen t to the cu rve x(t) = tpt (x) a t x = <p0(x).

A s u su a l , t h e s e no tions a r e lo c a l when r e s t r i c t e d to a neighbourhood of a point of M.

b) P r o p e r t i e s : co n v e rse ly :P ro p o s i t io n 1: Let X b e a v e c to r field on a m an ifo ld M; for ev e ry point x 0 of M, th e re e x i s ts a neighbourhood U of x Q, a r e a l n u m b e r e > 0 , and a lo c a l 1 - p a r a m e t e r g roup of loca l t r a n s f o r m a t io n s tpt : U -» M, | t | < e, which in d u c es the g iven X.

T he proof i s g iven by so lv ing a d i f fe ren t ia l equation x(t) = X , w here x (t) = %(х о) and the in i t ia l condition x 0 = <p0(xo).

A t r a n s f o r m a t io n tp of M induces an a u to m o rp h ism tp, of the L ie a lg e b r a Jf(M) of v e c to r f ie lds on M by

( ? . X )x = (<P.)y <Xy) w h e r e <p(y) = x a n d X e ^ M )

it induces a l so an a u to m o rp h ism <p* of the a lg e b ra of d if fe re n t ia l f o rm son M, and, in p a r t i c u l a r , a n a u to m o rp h ism of the a lg e b ra .^(M) of functions on M:

(<p*f)(x) = f(<p(x)), f e ^ M ) , x e M

The a u to m o r p h is m s tp, and tp" a r e r e l a te d by

v>*((<p„X)f) =X(<p*f) f e ^ M ) , x e ^ M )

P ro p o s i t io n 2: Let tp be a t r a n s f o r m a t io n of M; if a v e c to r f ield X g e n e r a te s a lo c a l 1 - p a r a m e t e r g roup of lo c a l t r a n s f o r m a t io n s tpt, th e n the v e c to r field tptX g e n e r a te s tpo tpt о tp'1.

C o r o l la r y 2 : A v e c to r f ie ld X is in v a r ia n t by tp (i. e . tptX = X) i f and only i f tp c o m m u te s with tpt .

R e m a r k : T h e s e c o n s id e ra t io n s p e r m i t a g e o m e t r ic a l in te r p re ta t io n of the b r a c k e t of two v e c to r f ie ld s , which can a l so b e used a s a definition [ 16] :

P ro p o s i t io n 3: Let X and Y b e two v e c to r f ie lds on M. If X g e n e r a te s a lo c a l 1 - p a r a m e t e r g roup of loca l t r a n s f o r m a t io n s tpt , th e n the b r a c k e t Z = [X ,Y ] i s th e v e c to r field:

IA EA -SM R -11/18 99

Z — lim t - о (i

Y (Y - ( ^ ) .Y )>)m o r e p r e c i s e ly , a t x e M ,

Z,X l im t 0

x - ( ( Ы у )х)] = 0>)

1 .2 . L ie groups

a) D ef in i t ions : A L ie g roup is a g roup G which i s a l so a d if fe ren t iab le m anifo ld and which s a t i s f i e s th e following condition: the group opera t ion

i s a d if fe ren t ia b le m apping of G X G into G.We denote by La ( r e s p . Ra) the left ( r e s p . r i g h t ) t r a n s l a t i o n of G by an

e le m e n t a of G, i . e .

F o r a e G , ad a i s the in n e r a u to m o rp h ism of G defined by (ad a)x = a x a " 1, VxeG; th e se t of a l l su c h a u to m o rp h ism s is a g roup denoted by ad(G ) and c a l le d the adjoin t g roup of G.

b) Action of a Lie g roup on a v e c to r f i e ld . We suppose tha t X i s a v e c to r f ield on G; we say tha t X i s left in v a r ian t ( r e s p . r ig h t in v a r i a n t ) if it i s in v a r ia n t by a l l left t r a n s l a t i o n s L a ( r e s p . r igh t t r a n s l a t i o n s Ra), a e G . We define the L ie a lg e b r a g of G to be the se t of a l l le f t in v a r ian t v e c to r f ie lds on G, with the u su a l addition, s c a l a r m u l t ip l ic a t io n and b ra c k e t o pera t ion ; a s a v e c to r space , g i s i s o m o rp h ic with th e tangen t space Te(G) a t the iden ti ty , the i s o m o r p h i s m being given by the m apping which sends X e g into Xe (Xe is the va lue of X at e); th u s , n be ing the d im ens ion of G, g i s a su b a lg e b ra of d im e n s io n n of the L ie a lg e b ra of v e c to r f i e l d s 8C (G ).

E v e r y A e g g e n e r a te s a (global) 1 - p a r a m e t e r g roup of t r a n s f o r m a t io n s of G [ 14] . Indeed, i f <pt is a 1- p a r a m e t e r g roup of loca l t r a n s f o r m a t io n s g e n e ra te d by A and if <pt(e) i s defined fo r | t | < e , then tpt(a.) can be defined fo r j 11 < e fo r e v e ry a e G and i s equa l to L a (<?t(e)) as <pt co m m u te s with e v e ry La; s in c e <pt (a) i s defined fo r |t | < e fo r a l l a e G , tpt(e) i s defined for | t | < oo and so i s VaeG; set

G X G -----►G

(a, b) I-----► ab"1

La x = ax ( re sp . Ra x = xa), VxeG

a t = <Pt(e )

then

a. = a, a V ( s , t ) e E X IR;t+s t s '

we ca l l a ( th e 1 - p a r a m e te r subgroup of G g e n e ra te d by A.

1 0 0 DOLBEAULT

A n o th e r c h a r a c t e r iz a t io n of a t i s tha t th e re is a unique cu rve in G such th a t i t s tangen t v e c to r â t at a t is equal to L a tAe, and a0 = e; in o th e r w ords , a i s th e unique so lu tion of the d if fe ren t ia l equation

with in i t ia l condition

Denote a 1 = <рг(е) by exp A; it follows tha t

exp tA = a ( VteIR

The m app ing A -» exp A of g in to G i s ca l le d the exponentia l m apping and we have:

<Pt (x) = x a VxeG

Let срЪе an a u to m o rp h ism of the Lie g roup G; it induces an a u to m o rp h ism cp, of the L ie a lg e b ra g of G with the following p r o p e r t i e s : i f A i s an e lem en t of g, then <p„A i s a le ft in v a r ian t v e c to r field; if A and В a r e two e le m e n ts of g, then

<p, [ A , В ] = [ cpt A , % В ]

In p a r t i c u la r , for e v e ry a£ G , the au to m o rp h ism ad a which m a p s x£G into a x a " 1 induces an au to m o rp h ism of g, a l so denoted ad a . T he r e p r e s e n ta tion a -» ad a of G i s ca l le d th e adjoin t r e p r e s e n ta t io n of G in g. F o r e v e ry a e G and A e g , we have (ad a) A = (Ra-1)eA, b e c a u s e A i s left in v a r ia n t .

L e t A and В be e le m e n ts of g and le t <pt be the 1 - p a r a m e te r g roup of t r a n s f o r m a t io n s of G g e n e ra te d by A; we obta in the following in te r p re ta t io n of the b ra c k e t :

[B , A] = l im 0 - ((<pt В - B)^) = l im (^- ((ad а,.1) В - B)

T h is follows f ro m p ro p o s i t io n 3 and f ro m the fact tha t g is a se t of left in v a r ia n t v e c to r f ie ld s .

c) Action of a L ie g roup on d if fe re n t ia l f o r m s . A d if fe ren t ia l fo rm a is s a id left in v a r ian t if

( L a ) '# = a, VaeG.

T h e se t of a l l left in v a r ia n t 1 - fo r m on G is a v e c to r space g” which is the dual sp a c e of g ( i . e . for A e g and n e g ’ , the function o(A) i s constan t on G).

L e t do be the e x t e r io r d if fe ren t ia l of the d if fe ren t ia l fo rm a; if a is le f t - in v a r i a n t , so i s a l so da, and, in p a r t i c u la r , we have

aeg’do (А, В) + I о ([ A, B] ) = 0 j

A, B e g

IA EA -SM R -11/18 1 0 1

T h is equa li ty i s known a s th e M a u r e r - C a r t a n equation [12 , 16] .Definition: The ca n o n ic a l 1 - fo r m в on G i s the le f t - in v a r ia n t , g -va lued1 - fo rm sa t is fy in g to

0 (A) = A VAeg.

T h is condition d e t e r m in e s в uniquely .It is often usefu l to have an e x p r e s s io n of в in lo c a l c o - o r d in a t e s .

Let e j , . . . , en be a b a s i s fo r g; we can se tn

в = ^ e.i = 1

we in t ro d u c e the s t r u c t u r e co n s tan ts Cl, by se t t ing " jk

i = 1

then th e M a u r e r - C a r t a n equation ta k e s the fo rm :

n

d e1 + i c j k 6j Л 0k =0 (i = 1, . . . , n)

j .k = l

R e m a r k : the s e t { : is a b a s i s fo r g°.

d) L ie t r a n s f o r m a t io n s g roups [ 14] :Definition: We say tha t a L ie g roup G i s a L ie t r a n s f o r m a t io n s g roup on a m anifo ld M, o r th a t G a c ts (d iffe ren tiab ly) on M if the following conditions a r e sa t is f ied :(1) e v e ry a e G induces a t r a n s f o r m a t io n of M

x ---- » x a VxeM

(2) th e m apping

GX M ----- » M

(a, x) I ► xa

i s a d if fe ren t ia b le m apping .

r VxeM(3) x(ab) = (xa) b, J

[ V(a, b )£ G X G

We a lso w r i te Ra x for xa and say th a t G a c ts on M on the r ig h t .We say tha t G a c t s effec tive ly ( r e s p . f r e e ly ) on M if Ra x = x for a l l

x e M ( re s p . som e xeM ) im p l ie s tha t a = e .Suppose tha t G a c ts on M on the r ig h t , c o n s id e r an e lem en t A of g, and

a s s o c ia te to A, for t r e a l , the set

a ( = exp t A

102 DOLBEAULT

T h is se t i s a 1 - p a r a m e t e r subg roup of G and i t s ac tion on M induces a v e c to r f ie ld A*. We have the p ropos i t ion :P ro p o s i t io n 4 : Let a L ie g roup G a c t on the m anifo ld M on the r ig h t . The mapping:

a : g ---- ► SC ( M)

A l---- ►A'

is a L ie a lg e b r a h o m o m o rp h ism .If G a c ts e f fec tive ly on M, then a is an is o m o rp h is m of g into S f { M).If G a c ts f re e ly on M, then , fo r ea ch n o n - z e ro A e g , a (A) n e v e r v an ish es on M.

2. CONNECTIONS IN A P R IN C IP A L F IB R E BUNDLE

2 . 1 . S ubspaces of tangen t sp a c e s

L et P = P(M, G) be a p r in c ip a l f ib re bundle on a m an ifo ld M, with s t r u c t u r a l g roup G and p ro je c t io n p; fo r u e P , le t Fu be the f ib re th ro u g h u. C o n s id e r the tangen t bundle T (P ) of P and the s e t TU(FU) of v e c to r s tangent a t u to the f ib re F u. Suppose tha t we can choose in T U(P) a se t TU(P) of v e c to r s sa t is fy in g the following conditions:

(a) TU(P) i s th e d i r e c t sum of TU(FU) and TU(P).

TU(P) = TU(FU) + T „(P)

(b) The c o r re sp o n d e n c e u -» TU(P) i s in v a r ia n t by G. A s G a c ts on P by r ig h t t r a n s l a t i o n R a u = ua, a e G , th i s can be w ri t ten :

T ua(P) = (Ra)„Tu(P), VueG, VaeG.

(c) TU(P) depends d if fe ren t iab ly on u.When such a d ec om pos i t ion i s p o ss ib le , we sha ll say:

Definition: Tu(-Fu) i s the v e r t i c a l su b sp ace of TU(P) at u; TU(P) i s a h o r izo n ta l su b sp a ce of TU(P) at u.

Given TU(FU) and TU(P), e v e ry v e c to r X ue T u(P) a d m its a unique decom posi t ion :

Xu = 'Xu + b l

w h ere I X u e T u ( F u ) and X u e T u ( P ) .

|XU and X u a r e , r e s p e c t iv e ly , the v e r t i c a l and h o r iz o n ta l com ponents of Xu.To have r e s u l t s about ex is te n ce of f ie lds of h o r izo n ta l v e c to r su b sp a c e s ,

le t us c o n s id e r a su b se t N of M; suppose we can define a f ield of h o r izo n ta l v e c to r sp a c e s on N by a s s o c ia t in g to e v e ry u e P , such tha t p (u)eN , a h o r iz o n ta l v e c to r su b sp a ce TU(P) of TU(P). We have th e following:

T h e o r e m 5: If M i s p a ra c o m p a c t , e v e ry f ie ld of h o r iz o n ta l v e c to r sp a c e s on the r e s t r i c t i o n of P to a c lo se d su b se t of M can be ex tended in a f ie ld of h o r iz o n ta l v e c to r sp a c e s on P .If M i s co m p ac t , P a d m i ts a f ie ld of h o r izo n ta l v e c to r sp a c e s .

P r o o f i s b a s e d on th e notion of p a r t i t io n of unity.

IA EA -SM R-11/18 103

a) D ef in i t ions : If we can define a f ie ld of h o r iz o n ta l su b s p a c e s in T(P)we say tha t we have a connect ion T i n P .

G iven a connect ion T i n P , we a s s o c ia te with it a 1 - fo r m и on P , with v a lu e s in g a s follows:

F o r ea ch v e c to r f ie ld X e T (P ) , we define w(X) to b e the unique A e g such tha t

A’ = IX.

The f o rm u i s c a l le d the connection f o rm of Г .

b) P r o p e r t i e s of the connect ion fo rm :

1) (a) VAeg, u(A* ) = A; in p a r t i c u l a r u(X) = 0 < = > X i s h o r iz o n ta l .Ф) F o r ev e ry a e G and ev e ry v e c to r f ie ld X on P , u ((R a)eX) =

a d ía "1) и (X).P r o o f : P r o p e r t y (a) follows d i r e c t ly f ro m the defin ition of u .To p rove (13), one m a y c o n s id e r e v e ry v e c to r f ield X a s the sum of IX and X and study s e p a ra te ly ea ch of th e s e two com ponen ts .

2) C o n v e rs e ly , le t и b e a 1 - fo r m on P , s a t is fy in g (a) and (0); then u define a unique connection Г о п P for which the connection f o rm is u .P r o o f : T h is p ro p e r ty follows d i r e c t ly f ro m th e r e la t io n be tw e en f o rm s and v e c to r f ie ld s .

3) T r a n s la t in g t h e o r e m 5 in t e r m s of connect ions , we have the following p ro p o s i t io n about ex is te n c e of connections:P ro p o s i t io n 6: И N i s a c lo sed su b se t of a p a ra c o m p a c t m an ifo ld M, everyconnec t ion on th e r e s t r i c t i o n of P(M) to N can be ex tended in a connectionon P(M ).If M i s com pac t , P(M) a d m its a connection .

4) If P(M) a d m its a connection, we sh a l l see l a t e r ( 2 .4 , a, exam ple) tha t it a d m its o th e r connec t ions .

5) L oca l c o - o r d in a te s : L et {Ua} be an open cove r ing of M, with a fam ily of i s o m o r p h i s m s

: P ' 1(Ua ) ---- ~ U a X G

and the c o r re s p o n d in g fam ily of t r a n s i t io n functions:

ф , : U П U„ ---- » G

F o r ea c h a, le t sa : Ua -* P b e the c r o s s - s e c t i o n on Ua , defined by

2 . 2 . Connections

-x \ XGU«’s a(x ) = |//â (x, e)

e i s the identity of G

104 DOLBEAULT

L et в be th e canon ica l 1 - fo r m on G. In a n a tu ra l way, we a s s o c ia te to в a g -v a lu e d 1 - fo rm on Ua П Ug / <¡¡ by se t t ing

ваВ = < 6 9

fo r ea ch a, define a g -v a lu e d 1 - fo rm ua on Ua by se tt ing

ioa = s„ u

then we have:

P ro p o s i t io n 7: On Ua П Ug, th e fo rm s Qa g and a r e sub jec t to the conditions:

ug = ad (фа g ) ioa + вав

C o n v e rs e ly , g iven the canon ica l fo rm в and a fam ily of g -v a lu e d 1 - fo rm s {toa }, ea ch defined on UH and sa t is fy in g the p re c e d e n t condition on Ua П Uj,then , th e r e i s a unique connect ion fo rm ш on P which g ives r i s e to{u)K} inthe way d e s c r ib e d above.

2 .3 . P a r a l l e l i s m

a) Lift of a v e c to r f ie ld on M :

T h e p ro je c t io n p : P ---- ► M induces a l in e a r mapping

P • (? ) T p(u) (M)

when a connect ion Г i s g iven on P , the p ro je c t io n p is an i s o m o rp h is mbetw een Тц (Р) and T p(u) (M). C onve rse ly :Definition: Given a v e c to r f ie ld X on M, th e r e e x i s t s a unique v e c to r field2 on P , w hich i s h o r iz o n ta l and which p r o je c ts onto X . T h is v e c to r f ie ld X i s ca l le d the h o r iz o n ta l lif t (or s im p ly the lift) of X.

R e m a r k : The ex is te n c e and un iqueness follow f ro m the fac t tha t p i s a l in e a r i s o m o rp h i s m of TU(P) onto T p(U) (M).

P r o p e r t i e s :- The s e t of l if ts of a l l v e c to r f ie ld s on M is in v a r ia n t by the r igh t

t r a n s l a t i o n s of G.- F o r two v e c to r f ie lds X and Y on M:

[ X, Y ] = [X, Y]

b) Lift of a c u rv e on M

B y c u r v e с = {x(t), 0 S t s 1} in M, we sh a l l m ean a m ap

IA EA -SM R -11/18 105

which i s p ie ce w ise d if fe ren t ia b le of c l a s s C1. A h o r iz o n ta l lif t (or s im p ly alif t) of С i s a cu rv e g = {u(t) , 0 s t a 1} in P which is:

с i s p ie ce w ise d if fe ren t iab le of c l a s s C 1,

с p r o je c ts on c: p(u(t)) = x(t) V t e [ 0 , 1],

e v e ry v e c to r tangen t to с i s h o r iz o n ta l .

E x is te n c e and u n ic i ty : g iven a cu rv e с in M, and a point u j E P in the f ib re o v e r x(0); p(u0) = x(0), t h e r e e x is ts a unique lift ç of c, s t a r t in g f ro m u0. (This i s nothing bu t the e x is te n ce and un ic ity of the so lu tion of a l in e a r d if fe ren t ia l equation with g iven in i t ia l condition).

Definition: C o n s id e r a cu rv e С and i t s lif t ç s t a r t in g f ro m u0 G P; such a lif t ends at иг such th a t p(uj) = xj. We thus ob ta in a c o r re sp o n d e n c e betw een u Q and Up we sh a l l w r i te i t c, i . e .

C ( U 0 ) = U j

L e t u0 d e s c r ib e the f ib re p"1(x0); we thus ob ta in a m apping с of the f ib re p _1(x0) onto the f ib re p ' ^ x j . T h is m app ing с i s , in fac t, an i s o m o rp h i s m and i s ca l le d the p a r a l l e l d isp la c e m e n t a long the cu rve c.

R e m a r k : E v e r y r ig h t t r a n s l a t i o n of G m a p s a h o r izo n ta l cu rve onto a h o r iz o n ta l cu rv e , so tha t R a co m m u te s with c:

R a о с = с o Ra; V a e G .

It fo llow s tha t с i s an i s o m o rp h is m be tw een two f ib re s of P .

2 .4 . C u rv a tu re and to r s io n

a) T e n s o r ia l f o rm s . Let p be a r e p r e s e n ta t io n of the s t r u c t u r e g roup G on a f in ite d im e n s io n a l v e c to r space V (i. e . p(a) i s a l in e a r t r a n s f o r m a t io n of V f o r each e le m e n t a of G and p(ab) = p(a) p(b) fo r each s e t (a ,b ) of e l e m e n ts of G.)

Definition: A t e n s o r i a l fo rm of d e g re e r , of type (p,V) on P i s a V -va lued r - f o r m s o n P such tha t:

(i) Ra* a = p (a -1) • a V a e G(ii) a (X j , . . . , X r ) = 0 w h enever one of the v e c to r s X j , . . . , Xr i s

tan g en t to a f ib re .In p a r t i c u la r , if p i s th e ad jo in t r e p r e s e n ta t io n of G in g, a fo rm a

sa t is fy in g (i) and (ii) i s a t e n s o r i a l fo rm of type ad G .

E x am p le : Let и be a connect ion fo rm on P and a be a t e n s o r i a l 1 - fo r m oftype ad G on P; then to = to + a defines a f ie ld of su b sp a c e s of T (P ) which can be taken as a h o r iz o n ta l v e c to r f ield , so tha t u defines an o th e r connection on P [ 1 3 ] . C o n v e rse ly , i f to and u a r e two connection f o rm s on P , t h e i rd i f fe re n c e i s a t e n s o r i a l 1 - fo r m of type ad G.

106 DOLBEAULT

R e m a r k : Let Г be a connection on P and in a point uE P , le t h be the p ro je c t io n

h : TU( P ) ---- »- TU(P)

Let a b e a V -v a lu e d r - f o r m on P sa t is fy in g only (i); we a s s o c ia te to a a t e n s o r i a l r - f o r m of type (p, V), by se tt ing

( «h ) ( X j .........X r) = e (Xi, . . . . Xr)

b ) E x t e r i o r c o v a r ian t d i f fe re n t ia t io n . The r e m a r k above im p l ie s :

Definition: g iven a connection F i n P , we can a s s o c ia te to the o p e r a to r d ofe x t e r io r d if fe ren t ia t io n an o p e r a to r V defined, fo r ev e ry f o rm a on P , by

Va = (da) h

which i s ca l le d o p e r a to r of c o v a r ian t e x t e r io r d if fe ren t ia t io n .T h e f o rm Va i s ca l le d e x t e r io r c o v a r ian t d e r iv a t iv e of a. T h is

def in i t ion im p l ie s :

P ro p o s i t io n 7: Given a V -va lued r - f o r m on P , sa t is fy ing (i), i t s e x t e r io r co v a r ia n t d e r iv a t iv e i s a t e n s o r i a l ( r+ l ) - f o r m of type (p, V). In p a r t i c u la r , i f и i s the connect ion fo rm , Vu i s a t e n s o r i a l 2 - fo r m of type ad G.

c) C u rv a tu re and to r s io n of a connection.

D efin it ion : Given P with the canon ica l f o rm 9 and a connection fo rm u, we se t:

Г2 = Vu, © = V0

Í2 i s th e c u r v a tu re f o rm of the connection;© i s th e to r s io n fo rm of the connection .

P r o p e r t i e s of the c u r v a tu r e fo rm :S t ru c tu r e equation of E . C a rtan :

du = - j [u , со] + fî

P r o o f : At u e P , and fo r X u, Yu e'IJ1(P), w r i te the s t r u c t u r e equation:

d u (Хц, Yu) = - I [ u ( ^ ) , «(% ,)] + Y„)

both s id e s of th i s equa li ty a r e b i l in e a r and s k e w - s y m m e t r i c in Xu and Yu . Since e v e ry v e c to r of T(P) i s the sum of a v e r t i c a l v e c to r and a h o r izo n ta l v e c to r , i t i s suff ic ien t to v e r i f y the l a s t equation in th e t h r e e following c a s e s

1) Xu and Yu a r e ho r izo n ta l : then u (Хц) and u (Yu) a r e null and th e s t r u c t u r e equation r e d u c e s to the definition of

2) Xu and Yu a r e v e r t i c a l : then f2(Xu, Yu) = 0 s in c e Г2 i s a t e n s o r i a l fo rm and the s t r u c t u r e equation , analogous to th e M a u r e r - C a r t a n equation r e s u l t s of the defin ition of e x t e r io r de r iva t ion .

IA EA -SM R -11/18 107

3) Xu i s h o r iz o n ta l and Yu is v e r t i c a l : fo r the sa m e r e a s o n S7(XU, Yu) = 0; a s in the p re c e d e n t c a s e , i t r e m a in s , by com ing back to v e c to r f ie lds X and Y:

2 d u (X, Y) + u [X , Y] = 0

th is equa tion i s s im i l a r to th a t of M a u r e r - C a r t a n and can be p ro v e d in the s a m e way.

L oca l c o - o r d i n a t e s . With the no ta tions of s e c t io n 1, se t t ing

n n

u = ^ w1 e¡ , Í2 = ^ Q1 e¡i=i i — l

the s t r u c t u r e equation can be e x p r e s s e d a s follows:

n

d u 1 = - I ^ c jkuJAuk + ft1 (i= 1,. . . , n) j,k =l

B ia n c h i1 s id e n t i ty : V Í2 = 0.

P r o o f : B y the defin ition of V, it su f f ice s to p rove th a t df2(X, Y, Z) = 0 w h en e v e r X, Y and Z a r e a l l h o r iz o n ta l , and th is follows im m e d ia te ly f ro m the e x t e r io r d if fe ren t ia t ion of the s t r u c t u r e equation.

2 . 5 . L in e a r and aff ine connections

T h e s e a r e two p a r t i c u l a r , bu t v e r y im p o r ta n t c a s e s .

a) L in e a r co n n e c t io n . L e t G = GI (n; IR) and P b e th e bundle L(M) of l in e a r f r a m e s o v e r the m an ifo ld M of d im ens ion n. T hen th e canon ica l fo rm в i s a t e n s o r i a l 1 - fo r m of type (GI (n; IR), ]Rn). A connect ion in P i s then ca l le d a l in e a r connect ion on M. We have th e n the s t r u c t u r e equa tions of E . C a r tan :

d0 = - и л 0 + 0 , du = - u / \ u + f2

and B ia n c h i1 s id e n t i t ie s :

V © = Í2 а б , V Q = 0

b) Affine c o n n e c t io n s . L et An be th e r e a l affine sp a c e of d im ens ion n. The sp a c e tangen t to M at x can be c o n s id e re d a s an affine sp a ce AX(M); an aff ine f r a m e of M at x c o n s i s t s of a point f of AX(M) and a l i n e a r f r a m e at x We c o n s id e r the s e t A(M) of a l l aff ine f r a m e s of M and define a p ro jec t io n

p : A(M) ---- - M

by se t t in g

p(u) = x

108 DOLBEAULT

for e v e ry affine f r a m e u at x. It is well known tha t A(M) can be co n s id e re d a s a p r in c ip a l bundle with s t r u c t u r a l g roup A(n; IR) (the g roup of affine t r a n s f o r m a t io n s of An).

An affine connect ion Г of M is a connection in A(M).

c) R e la t ion be tw een l in e a r and affine connect ions on M . L(M) can be c o n s id e r e d a s a subbundle of A(M); nam ely ,

<p: A(M) -----► L(M)

m a p s u = (f ; X j , . . . , Xn ) into (Xj, . . . , X n), and tp i s a h o m o m o rp h ism .

P ro p o s i t io n 8: T h e r e e x i s t s a h o m o m o rp h ism

ip: A(M) ---- ► L(M)

T h is h o m o m o rp h is m m aps ev e ry affine connection T o f M in to a l in e a r connection Г of M. The c o r re sp o n d e n c e Г-» T b e tw e e n the se t of affine connect ions and the se t of l in e a r connect ions of the sam e m anifo ld M is 1-1 .

T h is p ropos i t ion ju s t i f ie s the fact th a t one of the w ords " l in e a r connect ion" and "a ff ine connection" i s of ten u sed in p lace of the o th e r .

R e m a r k : C om plex l i n e a r and affine connect ions can be defined in the sa m e way by r e p la c in g IR by (C.

2 .6 . M e tr ic connect ions

a) D efin it ions: Let E be the v e c to r bundle, a s s o c ia te d with the p r in c ip a l bundle P , with s ta n d a rd f ib re F n (F = IR o r (C) and p ro je c t io n pE .

A f ib re m e t r i c in E c o n s is t s in giving in ea ch f ib re an in n e r p roduct g such th a t , i f x e M and i f ax(x), cr2(x) a r e c r o s s se c t io n s of P , then g(a1(x), cr2(x)) depends d if fe ren t ia b ly on x.

A connect ion in P i s a m e t r i c connection if the p a r a l l e l d isp lac em e n t of f ib re s of E p r e s e r v e s the f ib re m e t r i c g, th a t i s : fo r ev e ry curve с = { x t , 0 s t s l } in M, th e p a r a l l e l d isp lac em e n t p ’ X q) -* p '^X j) i s i s o m e t r i c ( i . e . p r e s e r v e s g).

b) P ro p o s i t io n 9 : If M is p a ra c o m p a c t , ev e ry v e c to r bundle o v e r M a d m i ts a f ib re m e t r i c g; to such a m e t r i c g c o r re s p o n d s a connection T in P .

P r o o f : The f i r s t p a r t r e s u l t s of a p a r t i t io n of unity; the second i s a co n sequence of the f i r s t and of p ro p o s i t io n 6.

c) E x a m p le s :

1) We r e c a l l tha t a R iem ann ian m e t r i c on M is a c o v a r ian t t e n s o r f ie ld of d e g re e 2 which i s p o s i t iv e defin ite:

g(X, X) g 0 V X e ^ ( M )

g(X, X) = 0 < = > X = 0

IA EA -SM R-11/18 109

g(X, Y) = g (Y, X) jvxe^-(M)[ v Y e á ' ( M )

so th a t g def ines an in n e r p ro d u c t in the tangen t space TX(M).The se t (M, g) i s ca lled a R iem ann ian m anifo ld .

P ro p o s i t io n 10: E v e r y R iem a n n ian m anifo ld ad m its a unique m e t r i c connection with v an ish ing to r s io n , which i s ca l le d th e R iem ann ian connection of M.

2) L e t M be a r e a l m anifo ld of even d im ens ion n = 2 m . G en e ra l iz in g the notion of com plex m anifo ld , we define on M an a lm o s t com plex s t r u c t u r e by giving a t e n s o r f ie ld J w hich i s , a t e v e ry point x e M ,a n en d o m o rp h ism of the tangen t sp a c e TX(M) such tha t J2 = - Identi ty . The se t (M, J ) i s an a lm o s t com plex m an ifo ld .

A H e r m i t ia n m e t r i c on an a lm o s t com plex m anifo ld M i s a R iem ann ian m e t r i c g in v a r ia n t by the a lm o s t com plex s t r u c t u r e J:

g(JX, JY) = g(X, Y)

for any v e c to r f ie lds X and Y.On an a lm o s t com plex m anifo ld , a com plex tangen t v e c to r Z x is sa id of

type (1, 0) ( r e s p . (0, 1 )) i f t h e r e i s a r e a l tangen t v e c to r X x such tha t

Z x = Xx - iJX x, ( r e s p . Z x = X x + iJX x)

A v e c to r f ie ld of type (1, 0) such tha t Zf i s h o lom orph ic fo r e v e ry lo c a l ly defined ho lo m o rp h ic function f i s a ho lom orph ic v e c to r f ield .

G iven an a lm o s t com plex m anifo ld M with H e rm i t ia n m e t r i c g, the fundam enta l 2 - f o r m Ф i s defined by

Ф(Х, Y) = g(X, JY)

for a l l v e c to r f ie ld s X and Y, so th a t $ is a l so in v a r ian t by J .A l in e a r (o r affine) connection on M is sa id a lm o s t com plex i f i t is a

connect ion in th e bundle C(M) of com plex l in e a r (or affine) f r a m e s on M.We have the following p ro p o s i t io n :

P ro p o s i t io n 11: Let M b e an a lm o s t com plex m anifo ld with m e t r i c g; if the R iem a n n ian connect ion defined by g i s a lm o s t com plex , then the fundam enta l2 - f o r m Ф is c lo sed .

and sy m m etr ic :

3. CONNECTIONS AND CHARACTERISTIC CLASSES

3 . 1 . W eil h o m o m o rp h is m

L et G b e a Lie g roup with L ie a lg e b r a g. We c o n s id e r m u l t i l in e a r s y m m e t r i c m app ings of o r d e r k:

f k : g X g X . . . X g ----► В

110 DOLBEAULT

in v a r ia n t by G, i . e . such tha t VaeG and V ^ , t 2, , . . , t k) e g X g X . . . X g,f R ((ad a ) tb (ad a ) t2, . . . , (ad a) tk ) = ^ ( t j , t 2, . . . , t k); l e t Sk(G) be the se to f such fk and

S(G) = ^ Sk (G)

k>0

S(G) i s a co m m u ta t iv e a lg e b r a when we define a m u lt ip l ica t ion in the following way: we f i r s t c o n s id e r f k e Sk(G) and fcG SC(G); we define f k + í e Sk+e (G) by

(ti» • • •. t k+c) = " (kT jyr X ‘ ' to<k)^ t o (k'1 4 ’ ‘ ‘ ' * to (k+c))о

w h e re the su m m atio n is taken o v e r a l l p e rm u ta t io n s a of (1, . . . , k + i ); we then c o n s id e r two e le m e n ts fk + & and fx+ fJ of S(G) and define

(fk + f c ) (f1 + fj ) = fkf‘ + Л -1 + f{ f* + f{ fJ

L e t P b e a p r in c ip a l f ib re bundle o v e r a m anifo ld M, with s t r u c t u r a l g roup G. Given a connect ion T i n P , le t Г2 be the c u r v a tu re fo rm and define, fo r fk G Sk(G), and X .G Тц(Р), (1 s i s 2k)

fk (П) (Xj, . . . , X2k) = — Y eo f 2k(í2(Xo(1), Xo(2)), . . . , Xo{2k)))О

w h e re the su m m a tio n is ta k e n o v er a l l p e rm u ta t io n s a of (1, . . . , 2k), and e 0 deno tes th e s ign of th e p e rm u ta t io n ст. With th e s e no ta t io n s , we have the following:

T h e o r e m 12: L et P b e a p r in c ip a l f ib re bundle o v e r a m anifo ld M, with group G and p ro je c t io n p; le t Г Ь е a connection in P and Г2 i t s c u rv a tu re fo rm ; then:(i) fo r ea ch fke S k(G), th e 2 k - fo rm fk (f2) on P p r o je c ts to a unique c lo sed 2 k - fo rm , s a y 7 k(ii), on M, i . e .

fk (í2) = p*(Tk (f2))

9 ki i) le t w(fk) b e the e lem en t of the de R ham cohomology group H (M; Ш.) defined by th e c lo sed 2 k - fo r m Yk(i3); th e n w(fk) i s independent of the choice of the connect ion Г and

w : S(G) ---- ► M*(M; K)

i s an a l g e b r a h o m o m o rp h is m which i s ca l le d the W eil h o m o m o rp h ism .

Definition: Let V be a v e c to r sp a ce on IR (or (C) and f 1, . . . , f r a b a s i s fo r the dual sp a c e of V. A m app ing ф: V -* К ( r e s p . (С) i s c a l le d a polynom ial function i f i t can be e x p r e s s e d as a po lynom ial of f 1, . . . , f r . (This concepti s ev idently independent of the choice of th e b a s is ) .

IA EA -SM R-11/18 111

L et E be a com plex v e c to r bundle o v e r M, with f ib re Cm and group Gl(m; <E), and le t P i t s a s s o c ia te d p r in c ip a l f ib re bundle . G iven a connection Г о п P and i t s c u r v a tu re f o rm Í2, we f i r s t define po lynom ial functions fQ, . . . , fm on th e L ie a lg e b r a gl(m; C) by:

det(XIm - 2 f c П)= I fk ^ ) * m' k0 £ k £ m

As a consequence of the p r o p e r t i e s of £7, they a r e in v a r ia n t by ad(Gl(m; (C)). By the p re c e d e n t t h e o re m , th e re ex is ts fo r each к (0 ë k s m) a unique c lo se d 2 к - f o r m Yk on M such tha t

P ' (T'k) = M n )

w h ere p : P -* M i s th e p ro je c t io n , so th a t we can w ri te :

det(Im ‘ г Ь = P*(1 + 7 i + • • • + 7m)

T h e o r e m 13: T he к - t h C he rn c l a s s ck(E) of a com plex v e c to r bundle E o v e r M i s r e p r e s e n t e d by the c lo sed 2 к - f o r m Yk .

R e m a r k : T he P o n try a g in c l a s s e s and E u l e r c l a s s e s can be ob ta ined in a s i m i l a r way by tak ing a r e a l v e c to r bundle o v e r the m an ifo ld M.

3 . 2 . Chern c l a s s e s

R E F E R E N C E S

[1 ] BOTT, R ., CHERN, S. S . , Hermitian vector bundles and the equidistribution of the zeroes o f their holomorphic sections, Acta Math. 114 (1965) 71.

[2 ] CARTAN, E ., Les groupes d ’holonomie des espaces généralisés, Acta Math. 48 (1926) 1.[3 ] CARTAN, E ., L'extension du calcu l tensoriel aux géom étries non-affines, Ann. Math. (2), 38 (1947) 1.[4 ] CARTAN, H . , Notions d'algèbre différentielle; application aux groupes de Lie e t aux variétés où

opère un groupe de Lie, Colloque de topologie de Bruxelles (1950) 15.[5 ] CARTAN, H . , La transgression dans un groupe de Lie, Colloque de topologie de Bruxelles (1950) 57.[6 ] CHERN, S .S . , Characteristic classes o f Herm itian manifolds. Ann. Math. (2) 47 (1946) 85.[7 ] CHERN, S .S ., D ifferential geom etry o f fiber bundles, Proc. Int. Congr. Math, (1950) 2^397.[8 ] EHRESMANN, C . , Sur la notion de connexion infinitésim ale dans un espace fibré e t sur les espaces à

connexion de Cartan. Nachr. Oest. Math. G es., (1949) 22.[9 ] EHRESMANN, C . , Les connexions infinitésim ales dans un espace fibré différentiable. Colloque de

topologie de Bruxelles (1950) 29.[10] HIRZEBRUCH, Topological Methods in Algebraic Geometry, 3rd enlarged edition, Grundlehren der

Math. Wissenschaften 131, Springer-Verlag New-York (1966).[11] HUSEMOLLER, Fiber Bundles, M cGraw-Hill, New-York (1966).[12] KOBAYASHI, S. , NOMIZU, K . , Foundations o f D ifferential Geometry, Interscience tracts _15 1 (1963)

2 (1969).[13 ] LICHNEROW1CZ, A . , Théorie globale des connexions e t des groupes d'holonom ie. Edizioni Cremonese,

Roma (1955).[14] LICHNEROWICZ, A ., Géométrie des groupes de transformations, Dunod (1958).[15] MILNOR, J . , Characteristic classes mimeographed Notes, Princeton University, (1964).[16 ] PHAM MAU QUAN, Introduction à la géom étrie des variétés différentiables, Dunod (1969).

IAEA-S M R -11/19

INTRODUCTION TO GLOBAL CALCULUS OF VARIATIONS

H .I . EL IASS ON Mathematics Institute,University of Warwick,Coventry, Warks,United Kingdom

>Abstract

I NTRODUCTION T O GLOBAL CAL C U L U S OF VARIATIONS.In this paper, manifolds of maps, critical point theories and variation integrals are treated within the

framework of a global calculus of variations.

INTRODUCTION

We s h a l l be co n c e rn ed with m app ings of one m anifo ld S into ano ther m an ifo ld M. T hroughout th is p a p e r , we sh a l l a s s u m e S to be a com pac t, connected R iem ann ian m anifo ld of c l a s s C" and d im e n s io n n i 1, p o ss ib ly w ith boundary 3S. M sh a l l denote a co m p le te R iem ann ian m anifo ld of c la s s C“ and without boundary (having so m e f in ite d im ension) . TS -* S, TM -*■ M sh a l l denote the tangen t bundles of S, M and TsS, TXM the tangen t sp a c e s at s G S, x G M.

L e t L (TS, TM) - * S X M denote the C* v e c to r bundle with f ib re L (TsS, TXM) o v e r (s, x) G S X M. If f : S -* M is a Ck m a p , 0 â к S oo, we denote by f* TM the p u l l -b a ck of the tangent bundle of M by f, which is then a v e c to r bundle (of c l a s s Ck) o v er S with f ib re (f*TM)s = Tf , . M over sG S. S ect ions g in f*TM can a l so be in te r p r e te d a s " v e c to r f ie lds in TM along the m a p of f" , i. e. Ç : S-» TM such tha t ? (s) G T f,sj M fo r a l l s G S, as is u su a l ly found to be m o r e convenien t if f is a cu rv e (n = d im S= 1). If n= 1 and thus f is a c u rv e , then the d e r iv a t iv e 3f = (d /d s ) f ( s ) is a s e c t io n in f*TM. M o re o v e r , fo r n > l we can in te r p r e t the ( f ir s t) d e r iv a t iv e 9f of f as a s e c t io n in the v e c to r bundle L(TS, f*TM) -» S, which has L (TsS ,T f^ M) as a f ib re o v e r s G S, the d e r iv a t iv e a t s be ing a l in e a r m a p 9f(s) :TsS-* Tf^ M.

Now le t th e re be g iven a continuous function:

F : L (TS, TM) -*• E

T hen fo r any C1 m a p f : S-*M we can fo rm the in te g ra l :

J (f) = J F O fs

w h e re we in te g ra te with r e s p e c t to the R iem ann ian m e a s u r e on S. Let h : S M be a g iven C“ m a p and denote by C1(S, M ^ the s e t of f in C ^ S , M)

113

114 ELIASSON

with f(s) = h(s) fo r a l l s e 8S and f hom otopic to h (fixed boundary). Let f t : S -»■ M be a C 1 v a r ia t io n of f = f 0 keep ing the boundary of S f ixed, i. e.(t, s) -* ft (s) is a C 1 m a p : [ 0 , e ) X S - * M with ft (s) = f 0(s) (= h(s)) fo r s £ 3S a t t = 0. Then with

? ( s > = | f f t (s> l t - o e T f ( . ) M

? i s a f ie ld a long f, ca l led the c o r re sp o n d in g in f in i te s im a l v a r ia t io n of f (a v a r ia t io n ft with a g iven se c t io n f in f*TM a s i ts in f in i te s im a l v a r ia t io n could be g iven by f t (s) = exp (t f (s)), with exp: TM -» M the exponen tia l m ap of M). The a s s o c ia te d v a r ia t io n of J is g iven by:

dJ ( f ) - ? = ± J ( f t ) | t=0

d j (f) : t f t ^ 'T M lo ->• IR is a l in e a r m a p , the v a r ia t io n a l d e r iv a t iv e of J , and can in fac t be in te r p re te d a s the d e r iv a t iv e of J : (^(S.M)),-* IR. J is sa id to be s ta t io n a ry a t f o r f a c r i t i c a l point of J , iff d j( f ) = 0.

The b a s ic ques t ions in ca lc u lu s of v a r ia t io n s a re :I. If J is bounded below, does J a s s u m e its m in im u m ?

II. Does J have c r i t i c a l points and what is the s t r u c t u r e of the s e t of c r i t i c a l po in ts?

E x am p le 1. S is a bounded dom ain in IRn with a С boundary , M = ]R and

J(f) = | / | | Df(x) f dx = i f J ( - J 0 dxS S i = i

E x am p le 2. S = [0 ,1] С IR, M as above and

l

J(f) = \ J II 9f(s) (I2 9sо

Then, g iven h, the c r i t i c a l po in ts of J a r e exac t ly the g eo d e s ic s in M f ro m h(0) to h ( l ) and hom otopic to h. H ere || Эf ( s ) || is the n o rm of the tangent 3f(s) to the c u rv e f(s) in the R iem ann ian m e t r i c of M. J(f) is ca l le d the en e rg y of the cu rv e f.

E x am p le 3. S and M g e n e r a l a s above and

IAEA -S M R -11/19 115

the e n e rg y of f. H ere L(TsS, T XM) is given the m e t r i c :

n

<A. B> = £ < А ’ вр В - e¡> i = l

with {ej, . . . , en} som e o r th o n o rm a l b a s e of Ts S. The c r i t i c a l po in ts a r e the so -ca l le d h a rm o n ic m a p s of S into M. In fac t , we have h e r e a n a tu ra l g e n e ra l iz a t io n of both p ro b le m s in e x a m p le s 1 and 2.

T he m e th o d s u s e d to so lve the c l a s s i c a l D ir ic h le t p ro b le m cannot, it s e e m s , be ex tended to highly n o n - l in e a r (M a m anifold) p r o b le m s a s in ex a m p le s 2 and 3. The r e a s o n is tha t in the f i r s t p lace it is d ifficu lt to dea l with, o r even in tro d u c e , any u se fu l w ea k e r topology on C °(S ,M ) than the C° topology i t s e l f and second ly , should we su c ce ed in doing so , a sequence in a hom otopy c l a s s in C°(S,M), converg ing weakly , would p o ss ib ly not have a l im i t in the s a m e homotopy c l a s s . Now one would even find it d ifficult to so lve the c l a s s i c a l D ir ic h le t p ro b le m by c o n s t ru c t in g a m in im iz in g sequence conve rg ing in C°(instead of L2, which is easy ) . Thus it m ig h t not be s u r p r i s in g tha t so f a r v e r y l i t t le is known about the p ro b le m of h a rm o n ic m a p s (exam ple 3) in i ts full g en e ra l i ty . Not m u c h has been a c co m p lish e d on th is p ro b le m s in c e the w ork of E e l ls and S am pson [2] . T h is is in g r e a t c o n t ra s t w ith the s u c c e s s of m any a u th o rs in d ea l ing with the o n e -d im e n s io n a l ene rgy function in exam ple 2, u s ing the ex tens ion of P a la i s and S m ale of M o rse T h eo ry to in f in ite d im e n s io n s [13]. T h is s u c c e s s is so le ly due to the one- d im e n s io n a l i ty of S and b r e a k s down fo r n> 1 a s then J can no lo n g e r s a t is fy the b a s ic condition (C) of P a la i s and S m ale [13] .

H ow ever, som e s u c c e s s can be r e p o r te d in u s in g c r i t i c a l point th e o r ie s a s deve loped by P a la i s , Sm ale and o th e rs on v a r ia t io n p ro b le m s fo r n> 1, e s p e c ia l ly fo r s t ro n g ly e l l ip t ic in te g ran d s F of h igh d e g re e m > n /2 in f.[4 , 10, 15, 16].

In th is p a p e r , we sh a l l d e m o n s t r a te how c r i t i c a l point th e o r ie s can be app l ied to obta in the ex is te n c e of c r i t i c a l po in ts of a v a r ia t io n in te g r a l J.We sh a l l m a in ly be co n c e rn ed with the p r e p a ra t io n s fo r th e se ap p l ica t io n s , which c o n s is t of en la rg in g the dom ain of J , such tha t J b ec o m e s a d i f fe re n t ia b le function and e s ta b l i s h su ff ic ien t p r o p e r t i e s of J.

We sh a l l , in th is s h o r t p a p e r , only be ab le to b r in g so m e m a in f e a tu re s of the th e o ry and m u s t th e r e f o r e r e f e r to the l i t e r a tu r e for m o s t of the p ro o fs . In s tea d , we sh a l l w ork th rough a few i l lu s t r a t iv e exam ples .

1. MANIFOLDS OF MAPS

We sh a l l f i r s t endow the sp a c e C°(S, M) of continuous m a p s S -* M with a C" B anach m an ifo ld s t r u c t u r e by c o n s t ru c t in g a s p e c ia l a t la s . Let exp : TM -* M denote the exponen tia l m ap fo r M and expx the r e s t r i c t i o n of exp to the tangen t sp a c e T XM a t x in M. The d e r iv a t iv e of expx a t v € T xM is a l in e a r m ap :

D expx (v) : TXM - TexpvM

116 ELIASSON

and D expx(0) = iden tity is a w ell known p r o p e r ty of the exponen tia l m ap. Thus th e re is an open neighbourhood U of the z e r o s e c t io n (the s e t of z e r o v e c to r s ) in TM, such tha t ( t , exp) : U -» M X M is a C°° im bedding of U onto an open neighbourhood W of the d iagonal in M X M. H ere r : TM-> M is the p ro jec t io n .

Now le t h; S-* M be a C°° m ap . Then C°(h*TM) is a B anach sp a ce with n o rm :

h*U is an open neighbourhood of the z e r o se c t io n in h*TM and C°(h*U) is open in C°(h*TM) ( r e m e m b e r S is com pact.1 ). The m ap

T his m a p is e a s i ly se e n to be continuous and open and thus a h o m e o m o rp h ism onto i ts ima'ge in C°(S,M), which c o n s is t s of a l l f £ C ° ( S , M), such tha t (h, f ) :S - * W C M X M and we sh a l l denote by C°(h*W). In fac t , W-*M given by the p ro je c t io n (x ,y)-*x is a f ib re bundle and identify ing a m a p of f : S-> M with the c o r re sp o n d in g se c t io n (id, f) in h*W, C°(h*W) s im p ly denotes the sp a c e of continuous se c t io n s in h*W and the h o m e o rm o rp h is m

II? II = sup II I (s) II, ? e c ° ( h * T M )C° se s

exph : h * U - M; e XPh(? ) = e x p ^ ?, ? G Th(s) M

is in jec t ive on each f ib re and induces an in jec t ive m ap

C°(exph) : C°(h*U) -* C°(S,M)

C°(exph) (?) = expho ?

C°(exph) : C°(h*U) - C°(h*W)

is induced by the С f ib re bundle equivalence :

( V exph)

h!;;U - h*W

S S

Now le t h j, h 2 6 C “ (S,M) and suppose

C°(h*W) П C °(h | W) = C °(hfW П h |W ) f ф

T hen ea ch f ib re of W12 = h* W О h^W is a non em pty open su b se t of M and

IA EA -SM R-11/19 117

C°(exphi)-1(C°(W12)) = C°(U.)

with

u i = K j ' e x Ph¡) C h î U

The t r a n s f o r m a t io n of c o o rd in a te s of an f GC°(W12), f " exPhj°? exPh2or> is then induced by the d if feom orph ism :

In o r d e r to show tha t the co l lec tion C°(exph) of c h a r t s , with h in C°(S,M), is an a t l a s , we have to show tha t С°(Ф12) is a C ” d if fe o m o rp h ism . We sh a l l do th is in a m o r e g e n e r a l s e t t in g u s in g the m e thods in Ref. [3].

L et VB(S) denote the c a te g o ry of C ” v e c to r bundles o v er S. A sec t io n fu n c to r Г a s s ig n s to ea ch EG VB(S) a B anachab le sp a ce Г (E) conta in ing C°°(E) a s a d en se l in e a r su b sp a ce and ea ch m o r p h is m A 6 C " ( L ( E , F ) )E , F £ VB(N), the m o r p h is m A‘ g in Ц Г ( Е ) , F (F )) . Thus Г (Е ) is i so m o rp h to the com ple t ion of С " (E) in som e n o rm || • || (which is d e te rm in e d only up to equ iva lence of n o r m s ) and

ф12= K 2 > exPh/^K j- exPhl) :Ur U2

С^ехр,, ^ „ ( ^ ( e x p ) = С°(Ф12)

A - 1 r S cons t H € Hr , (A- Ç) (s) = A(s) • ?(s)

define fo r | G C (E):Let E E VB(N) and choose so m e R iem ann ian m e t r i c fo r E . Then we. p oo

Il € H o = SUP II € (s)с

s

118 ELIASSON

and obtain the B anach sp a ce s C°(E) and L P(E) by com ple t ing C°°(E) in those n o r m s . Changing the m e t r i c on E only changes the above n o r m s into equiv a len t n o r m s and as

l |A - | | | co S II a ||co II € ||co

l |A -? | |LpS II A II co II ? llLp

we have sec t io n fu n c to rs C° and Lp. If we choose addi t iona lly to the m e t r i c a R iem ann ian connection f o r E , we can d if fe re n t ia te a s e c t io n f in E с ova r ia n t lyk - t im e s to obtain the к - t h - o r d e r c o v a r ia n t d e r iv a t iv e VKf a s a s e c t io n inL k(TS, E) and then define (for к = 1, 2 , .........., láp< oo):

Il C llc„ ■ i II ’ 'с llc.i = О

/ л .1/р

( I (КМ/)i =0

к 1/P

(/¿llv'íf)S i =0

к рT h is g ives the s e c t io n fu n c to rs С and L

Definition. A se c t io n func to r Г on VB(S) is ca l le d a m anifo ld m o d e l , iff

(1) We have a continuous l in e a r inc lus ion

F (E )C C °(E ) , a l l E e V B ( S )

i . e .

III Ilc 0 s cons t |U | | r a l l ? e C ” (E)

(2) We have a continuous l in e a r inc lu s ion

r ( L ( E , F ) ) C L ( T ( E ) , r ( F ) ) , a l l E ,F G V B (S )

Il A- g II § cons t | |a || | | | | | A 6 C " ( L ( E , F ) ) a l l ? e c “ (E)

i. e.

IA EA -SM R-11/19 119

(3) L et E , F E VB(S), U C E open with ea c h f ib re Us n o n -em p ty and Ф: U-* F a C“ f ib re p r e s e r v i n g m ap. T hen we have an induced continuous map:

Г(Ф) : Г ( и ) -* r ( F ) , Г ( Ф ) ( | ) = в » ?

R e m a rk .

Note tha t (1) im p l ie s Г ( и )= Г (Е ) n c ° ( U ) is open in Г (Е )

L e m m a 1. With UCE, Ф: U~* F a s in (3) and Г a m an ifo ld m o d e l on VB(S), Г(Ф) is a C“ m a p and ОГ(Ф) = Г ( В 2Ф), w here

D2$: U - L (E , F)

is the C” f ib re p r e s e r v in g m a p defined by

D2$ ( ? ) . n = A ®(C+tn) |t=0 = D®,(Ç).n; ? GUS

r, G E s , s e s and Ф5 = Ф I us : Us -► Fs

P roo f : We have:

ф» n - ® ° l - (d 2® « |) . ( n - f ) = e ° ( n , ? M r ) - ç )

withl

в (v, u) = J (ОаФ (v + t(u - v)) - ОаФ ^)) dt о

в :U'e u ' - L (E, F)

a C“ f ib re p r e s e r v i n g m a p and U ' so m e convex open s u b se t of U. Then

II Г(Ф )(п)- Г(ф) (f) - Г (Б 2Ф)(?) • ( n - f ) | |г

S cons t II Г(0) (l,r¡) ||r II Г) - c II г

u s in g (2). Then a s в (v ,v ) = 0 and Г(0) is continuous by (3) we have a6 > 0 fo r any e> 0, such that

С - T) | |г < б = Ф II Г ( 0 ) ( f , Г)) Hr < €

120 ELIASSON

p ro v in g D F (4 ) = T (D 2 Ф), which is a continuous m a p f ro m F(U) into F ( L ( E ,F ) ) by (3). r ( L ( E , F ) ) is con t inuously included in L (T (E )) , T(F)) and thus Г(Ф)is C1. The r e s t follows by induction u s in g the fo rm u la fo r D F ( i ) ,

L e m m a 2. L e t k = 0 , 1 , . . . . and lS p < o o . Then С k is a m anifo ld m o d e l and L^ is a m anifo ld m o d e l if

к > — (n = d im S)P

P roo f : The p ro o f fo r Ck is e a sy , u s in g the "L e ibn iz ru le " :

V ( A • f ) = V A ® f + A • VÇ

P r o p e r ty (1) fo r L^, k > n /p , is ju s t the w ell known Sobolev im bedding th e o re m and the o th e rs follow f ro m tha t and the Leibn iz ru le . I should only r e m a r k th a t one p ro v e s the continuity of Г(Ф) m o s t ea s i ly by f i r s t showing tha t it m a p s bounded s e t s of Г (и ) to bounded s e t s in F (F ) and then u se e s t im a te s as in the p ro o f of L em m a 1.

T h e o r e m 1. L et Г be a m an ifo ld m o d e l on VB(S). Then th e r e is a well defined s e t of m a p s r ( S ,M ) C C ° ( S ,M ) , such tha t r ( S ,M ) is a C“ B anach m an ifo ld m ode l led on the B anach sp a c e s r (h * T M ) , h e C " (S,M), and

Г (ех р ь) = C : (exph) | r (h * U ) : | - exp of

a s a c h a r t .

P roo f : We have f (h * U ) C C°(h*U) and define r ( S , M ) as the union of theim a g e s of C°(exph) : F (h * U ) -* C°(S, M) when h ru n s th rough C" (S, M). L e m m a 1 g u a ra n te e s tha t r ( S , M) is w ell defined and change of co o rd in a te s is given by a C “ d if fe o m o rp h ism .

The m o s t im p o r ta n t m an ifo lds of m a p s for v a r ia t io n p ro b le m s a r e the Sobolev m an ifo lds L P(S,M ), k > n / p . We have shown in Ref. [4] how to c o n s t ru c t ca non ica l F i n s l e r s t r u c t u r e s on th e s e m an ifo lds f ro m R iem ann ian m e t r i c s on S and M. The tangen t sp a c e of L k(S,M) a t f is ju s t L£(f*TM) and the L jJ-norm of a s e c t io n f in f*TM is defined by the s a m e fo rm u la as b e f o r e , now u s in g the m e t r i c and connection on f*TM induced by the R iem ann ian s t r u c t u r e on M [3]. Thus we obtain a n o rm on ea ch tangen t sp a ce of L ^(S ,M ), which p ro v id es L P(S,M) with a L ipsch i tz F in s le r s t r u c t u r e and a C*° R iem ann ian m e t r i c if p = 2 [4] . E a c h com ponent of L^(S ,M ) b ec o m e s a m e t r i c sp a ce and we have shown it to be com ple te [4] .

E x am p le 4. S is o n e -d im en s io n a l , n= 1, say , e i th e r the to ru s ( c i rc le ) IR/Z or the un ite in te r v a l [0. 1] and then in e i th e r c a se p a r a m e t r i z e d by 0 S s S 1.Then H^S, M) is a C " m an ifo ld (H1 = L 2) a s !>■§■. F o r x e H^S, M) we have

t xh 1(s , m ) = h V t m )

IAEA-SM R-11 /19 121

and H 1(SJ M) is a C“ R iem ann ian H ilb e r t m anifo ld with the in n e r p ro d u c t of two tangen t v e c to r s f , r) e H1(x*TM) given by:

 , = < ? , n > = < C ,4 > + <VC) Vn>H 1 1 0 u

with

1

< € , , > 0 = J < 5 ( 8 ) , Г) ( s ) > d s0

We have shown [4, 5] tha t

H W f S . M ^ T M ) ЭН°(х^ М )

I 1H ^ S . M ) Э X

is a C°° v e c to r bundle with R iem ann ian m e t r i c g iven by •( , )>Q. M o re o v e r , tak ing the tangen t of cu rv e s x in H ^ S .M ) is a C “ se c t io n , x - Эх, in th is v e c to r bundle . We sh a l l com pute th is s e c t io n in lo c a l c h a r t s . L e t h : S — Mbe a C“ cu rv e . If f G H ^h^TM ) is the lo c a l r e p r e s e n t a t i v e (co -o rd in a tev e c to r ) of x G H 1(S, M), we have x = exp оf . Now le t К : T2M —TM be theconnect ion m a p of M, such th a t the co v a r ia n t d e r iv a t iv e of a f ield С iscom puted by

V | = K (3 f ) ; 3 f (s ) = TC (s, 1 )

T f : TS = S X IR - T 2M the tangen t of ?. Then with т : TM - M, тг: T2M - TM the p ro je c t io n s , (т: , Т т , K) : T2M - T M © TM © TM is a C” d if fe o m o rp h ism [3 ]. T hen we obtain:

dx = Tx • 1 = T exp oT f • 1

= T exp о (Tj, Т т , K) 1 о (tj , Т т , K) ° 9f

= V exp » (?, 0h, V f )

= (Vj exp о f ) • 9h + (V2 exp » ? ) • Vf

H ere we have defined the c o v a r ia n t d e r iv a t iv e of the exponen tia l m a p by

Vexp = T exp ° (Tj , T t ,K ) 1: T M © T M 0 T M - TM

and u se d the fac t tha t Vexp (v ,u , w) is b i l i n e a r in (u, w) on each f ib re [3] .

122 ELIASSON

M o re o v e r ,

V2 exp (v) ■ u = D2 exp (v) • u = -J¡- exp (v + tu) 11 = 0

= D expx(v) • u; v , u e T xM

Thus V2 exp (v) : TXM-* Texpv M is a l in e a r i s o m o rp h is m fo r v suffic ien tly s m a l l , in p a r t i c u l a r fo r v £ Ü C TM, we u se d e a r l i e r in defining the c h a r t s . Now Vj exp (v) : TXM-*- TeXpv M is a l in e a r m a p and we obtain C°° f ib re m a p s :

9 : U - L (TM, TM)

в (v) = (V2 exp (v)) 1 Vj exp (v)

0h : h*U - h*TM; 0h(v) = в (v) • 3h

Then we obtain

Эх = (D2 expo? ) • (Vf + 0h ° I )

T h is shows that

8h( i) = V? + 0h c i

is the c o - o r d in a te v e c to r of Эх, s ince the lo c a l t r iv ia l iz a t io n of the v e c to r bundle H (\ h 1(S,M)*TM) is ju s t g iven by:

hV u ) x H°(h*TM) Э ( ? , n ) - (D2e x p o ? ) - n

Now we e a s i ly s e e th a t Э is a C” se c t io n , a s V; H1(h*TM) -* H°(h*TM) i s a continuous l in e a r m a p and

H1(hsiiTM) С H°(h*TM)

is of c l a s s C°° by L e m m a 1.L e t u s a l s o com pute the m e t r i c on H ^ H ^ S , M)*TM) loca lly . Let

h e C " ( S , M ) , x= exp °? with ? e H x(h*U) a s b e fo re . L et Y, Z e H ° ( x * T M ) b e loca lly g iven by f , Ç, i. e.

IAEA-S M R-11/19 123

l

<Y ,Z> = J <D2exp (? (s)) * rj (s), D2exp (f (s)) * Ç (s)> ds о

l

= J <Gh(f(s)) * r¡ (s), Ç (s)> dsо

= < ( G h o Ç) -T) , Ç > n 0

with G : L(TM, TM) a ( f f ib re m a p ,

G(v) = D2exp (v)*D2exp(v)

Gh = h*G : h*U - L(h*TM, h*TM)

ThusH1(Gh) : h V u ) - H2(L(h*TM)h* TM))

is of c l a s s C” by L e m m a 1 and we have a continuous l in e a r inc lus ion

H1(L (h*TM, h*TM)) С L(H°(h*TM), H°(h*TM))

by:

I|a - ? | |0 s II a ||c o | | ? | | 0 s cons t II a II j | | ç | | 0

The m e t r i c is thus of c l a s s C" and the co r re sp o n d in g n o rm equivalen t to the H °-no rm in H°(h*TM) fo r a l l f e H ^ h 'U ) , if we choose U bounded and such that G(v) ёб > 0 fo r a l l v £ U .

Now it is im m e d ia te tha t the e n e rg y function

E(x) = I II Эх ||¡ = i <Эх, Sx>0

is a С” function on H ^S .M ). The d e r iv a t iv e of E ex tends the v a r ia t io n a l d e r iv a t iv e :

dE (x) •? = <Эх, Vf >0

If x is a c r i t i c a l point of E , i . e . x E H ^S, M) and dE(x) = 0, then x can be shown to be a C°° cu rv e and then (in c a s e of S = [0, 1 ] , we r e s t r i c t E to the subm anifo ld of c u rv e s having fixed in i t ia l and end points):

dE(x) • I = - <V9x, f >0

so dE(x) = 0 is equ iva len t to V9 x = 0 o r x is a g eo des ic . We s h a l l l a t e r take up the ques tion of the ex is te n c e of c r i t i c a l points .

Then we get:

124 ELIASSON

O ur m a in a im h e r e is to d is c u s s the condition (Condition (C) by P a la i s and Sm ale [13] ), w hich has been shown to be s t ro n g enough to m ake it p o s s ib le to g e n e ra l iz e M o rse T h eo ry and L u s te rn ic k - S c b n i re lm a n ca te g o ry th e o ry to infinite d im e n s io n s .

L et X be a B anach m anifo ld of c l a s s C2 a t l e a s t . A n o rm on a Banach sp a c e bundle E over X is a continuous function v-* || v|| on E , such tha t the r e s t r i c t i o n to each f ib re E x of E is an a d m is s ib le n o rm on the Banach sp a c e E x. Suppose we have a lo c a l t r iv ia l iz a t io n of E o v e r a c h a r t of X:

E Э 7Г-1 (U) * ф (U) XIE

77 í IX 3 U - Ф ( и ) с в

h e r e В, IE a r e B anach s p a c e s , cp a C2 d if fe o m o rp h ism and Ф a hom eom orphis: w ith Фх: Е х->-]Е given by Ф(у) = (cp(x), ®x(v)), v e E , , a l in e a r h o m e o m o rp h ism . L oca l ly the n o r m function is g iven by:

N (x .Ç ) = II Ф '^х, f )||

T he n o rm on E is s a id to be u n ifo rm , iff fo r any co ns tan t k> 1 and x 0Gcp(U), we have:

i N (x ,Ç) S N ( x 0, ?) S kN(x, f )

fo r x in so m e ne ighbourhood of x 0. Since N(x0, | ) is equ iva len t to the n o rm of IE, th is im p l ie s tha t a l l the n o rm s N(x,C) on IE, p a r a m e t r i z e d by x in a neighbourhood of x0, a r e equivalen t to the n o rm of IE. The n o rm is Said to be loca lly L ip sc h i tz , iff fo r any xfl S cp (U), we have:

|N(x.S) - N(x0, f ) | s c II x - x 0H N (x0,Ç)

fo r so m e co n s tan t с > 0 and x in so m e ne ighbourhood of x0. This p ro p e r ty is in v a r ia n t ag a in s t a change in lo c a l t r iv ia l iz a t io n , if E is of c l a s s C1 and im p l ie s u n ifo rm ity , a s then

N ( x , f ) â (с I x - x 0 I + l ) N ( x 0, €) s k N ( x 0, | )

N (x, f ) ê (1 - c H x - x0 II )N (x0, C) s 1 /к N(x(x0, Ç )

fo r any k> 1, if II x - Xq|] is s m a l l enough.

2. C R ITICA L POINT THEORIES

Definition. A F i n s l e r s t r u c t u r e on X is a u n ifo rm n o rm on TX. A F in s l e r m an ifo ld is a r e g u la r B anach m anifo ld to g e th e r with a F in s l e r s t r u c tu re .

IAEA-S M R -11/19 125

L et f be a C 1 r e a l -v a lu e d function on a C 1 F i n s l e r m anifo ld X. f s a t i s f ie s condition (C) of P a la i s and S m ale , iff any sequence of po in ts xm in X, such tha t f ( x m) is bounded and | |d f (x m)|| co n v e rg es to z e r o fo r m -’■<», has a conv e r g e n t su bsequence . T hen of c o u r s e , if X is com p le te , the subsequence c o n v e rg e s to a c r i t i c a l point of f. The m a in th e o re m of the g e n e ra l iz e d L u s te rn ik -S c h n i r e lm a n th e o ry of c r i t i c a l po in ts (J. T. S chw artz [14] and R. S. P a la i s [11] ) is:

T h e o r e m 2. Let X be a co m p le te C 2 F i n s l e r m anifo ld and f : X-* IR a C1 function bounded below and sa t is fy in g condition (C). T hen f a s s u m e s i ts m in im u m on ea ch com ponent of X and has a t l e a s t cat(M) c r i t i c a l poin ts a l to g e th e r .

H e re , cat(M) is the L u s te rn ik -S c h n i r e lm a n ca ta g o ry of M, i. e. the n u m b e r of c losed s u b se ts of M, c o n t ra c t ib le in M, needed to co v e r M.A good d e s c r ip t io n of th is th e o ry can be found in Ref. [12] , w ith p ro o f of the above th e o re m . We sh a l l now p r e s e n t a m e thod to e s ta b l i s h the im p o r ta n t condition (C), s e e Ref. [6] .

A B anach m an ifo ld X is ca l le d weak subm anifo ld of an o th e r B anach m anifo ld X 0 contain ing X as a su b se t , iff fo r any x 0e X0, th e re is a c h a r t 9 o : U0 -» cp0 (u0 ) С в0 fo r X Q with x Q e UQ and a B anach sp a ce В e B0 with the in c lu s io n being continuous, such tha t the r e s t r i c t i o n of cp0 to U = X £ UQ is a c h a r t cp fo r X:

X 0 Э ^ Фо(и о> c B o

и и иx d u ! ф (и) с в

ф ( и ) = ф 0 ( и ) п в

We sh a l l ca l l such a c h a r t a weak c h a r t fo r X (at x0). It follows that the inc lu s ion of X into X0 is continuous, but X is not n e c e s s a r i l y c lo sed in X 0. We sh a l l ca l l a F i n s l e r s t r u c t u r e on X loca l ly bounded with r e s p e c t to Xq, iff fo r any x 0e X Qand cons tan t L , th e r e is a cons tan t c, such tha t

N (x, f ) s с II Ç II

holds fo r the n o rm function in x e ф (U), || x || < L, ? e В ( || || denoting the n o rm in B).

Now le t X, X0 be B anach m an ifo ld s with X a weak subm anifo ld of X 0.A function f :X-*IR is ca l le d weakly p r o p e r with r e s p e c t to X 0, iff any su b se t A C X , such tha t f is bounded on A, is r e la t iv e ly co m pac t in X0 , f is ca l le d loca l ly bounding with r e s p e c t to X Q, iff fo r any x 0 in X0 and cons tan t K,th e re is a weak c h a r t fo r X a t x0 and a c o n s tan t L , such tha t

Il x II < L fo r a l l x e ф (U) with f(x) <K

126 ELIASSON

A С 1 function f :X->IR is ca l le d loca l ly c o e rc iv e with r e s p e c t to X Q, iff fo r any x0e X 0 and co n s tan t L , th e re is a weak c h a r t fo r X a t x 0 and c o n s tan ts e > 0 and > , such tha t (in lo c a l co -o rd in a te s ) :

1. (Df(y) - Df(x)) • (y - x) â e D y - x II - A || y - x ||0, fo r a l l x, y e tp (U) with| |x 0 < L, Il у II < L, o r equ iva len tly , if f is of c l a s s C2, su c h tha t

2. D2f(z) ( I , I ) S e II ? II - A II Ç ||0 , fo r a l l x e cp(U) with || x || < L and a l l Ç e B.H ere II II , II 10 denote the n o r m s in B, B 0.

T h e o r e m 3. Let X be a C2 F i n s l e r m anifo ld and f : Х-» E a C1 function. Suppose th e re is a C2 B anach m anifo ld X0 contain ing X a s a weak subm anifo ld such tha t the F i n s l e r s t r u c t u r e of X is loca lly bounded with r e s p e c t to X 0, f is weakly p r o p e r , loca lly bounding and loca lly c o e rc iv e with r e s p e c t to X 0. T hen f s a t i s f i e s condition (C).

P ro o f . L e t x m be a se quence in X, such th a t f (x m ) is bounded and

| |d f ( x m )|| -*■ 0 fo r m-ôo

T hen a s f is weakly p r o p e r , we can find a su b se q u en c e of x m converg ing in X0 to so m e x0e X 0. T hen we choose a weak c h a r t a t x 0 , su c h tha t the m e m b e r s of the su b sequence in the dom ain cp(U) a r e bounded in В and such tha t inequali ty 1 ho lds . Now, a s the F i n s l e r s t r u c t u r e is loca lly bounded with r e s p e c t to X 0 , we m a y a s s u m e N(xm , Ç) S с | | | | | and then, u s in g 1:

+ 7 .(Il d f(xm )||+ Il df(xe) Il ) c H x m- x j

Now a s x m is a Cauchy se quence in B 0 and || x m || i s bounded in B, it follows th a t x m is a Cauchy sequence in В and thus x 06cp(U) and x m -+x0 in B.

T h e o r e m 4. L e t Г be a m anifo ld m o d e l on VB(S). T hen the m an ifo ld T (S ,M ) f ro m th e o re m 1 is a weak subm anifo ld of C °(S ,M ). M o re o v e r , L^(S ,M ) fo r к > n / p is a F i n s l e r m anifo ld with the n o rm g iven be fo re (in s e c t io n 1) and the F in s l e r s t r u c t u r e is loca lly bounded with r e s p e c t to C °(S ,M ).

P ro o f . That r ( S , M ) is a weak subm anifo ld of C°(S, M) follows im m ed ia te ly f ro m the c o n s t ru c t io n of c h a r t s fo r F (S ,M ) by r e s t r i c t i n g the c h a r t s of C°(S,M). The r e g u la r i ty of the sp a ce r ( S ,M ) follows d i r e c t ly f ro m the r e g u la r i ty of C°(S ,M ), so l £(S,M) is a F i n s l e r m anifo ld . It is not d ifficult to show tha t the F in s l e r s t r u c t u r e of L P(S,M) is loca lly bounded with r e s p e c t to С °(S,M), u s in g the lo c a l f o rm u la s in Ref. [4]. We sh a l l r e s t r i c t o u r se lv e s h e r e to H 1(S ,M ),n= 1, u s in g the lo c a l fo rm u la :

IAEA -S M R -11/19 127

N ( x , | ) = K G ox) - ? , D 0

+ <(G ox)-V h ( x ) - f , Vh ( x ) - O 0)^

w h ere Vh(x) • f is the lo c a l r e p r e s e n ta t iv e of the c o v a r ia n t d e r iv a t iv e of the v e c to r f ie ld (V2e x p °x ) - f and can be com puted by s i m i l a r m e thods a s u se d in E x am p le 4 (see Ref. [5] ):

Vh ( x ) - ? = V ? + (Л1о х ) - | + (Ao. x ) - ( 0 h (x), I )

withA j : h*U - L(h*TM , h ’fTM)

A 0 : h * U - L 2(h*TM, h*TM)

f ib re p r e s e r v i n g C " m a p s and 3h(x) the lo c a l r e p r e s e n ta t iv e of the tangentЭх (given in E x am p le 4). Now le t L> 0 be g iven and x S H x(h>:!U) with || x ||j S L.Then

Il 9h(x) I I | | V x | |0 + | |e h o x | |0 S L + C =Lj

II v h ( ^ ) * € l l 0 á II l i o + C i H € II o + C 2L i H € II с о

á Const II f ||a

thus N(x, I ) S с II f Hj.

E x am p le 5.

T h e o re m 5. Let S be e i th e r the unite in te r v a l [0, 1] o r the c i r c le IR/Z and le t HX(S, M)c denote the subm anifo ld in H1(S,M) of cu rv e s hom otopic to a g iven C “ c u rv e с : S -► M (with fixed in i t ia l and end point, if S = [0, 1] ). Then the e n e rg y function E : H ^S, M)c->-IR f ro m exam ple 4 is weakly p r o p e r (with M com pac t in c a s e S = Ш/ 2£), loca l ly bounding and loca lly c o e rc iv e with r e s p e c t to C°(S,M) and thus s a t i s f i e s condition (C).

P ro o f . L e t E(x) й К fo r x € A С H'Vs, M)c , then

s

d M(x(s), x(t)) s j J К Эх(r ) (I d r I t

s s

Ш I ^ l 2 d T ^ | ^ | | 9 x ( t ) | | d r | ^ t t

S I s - t I* (2K)i

128 ELIASSON

Thus A is an equicontinuous su b se t of C°(S, M). A(S) = {x (s ) | x E A, s E S } is r e la t iv e ly co m p ac t in M, s ince M is co m p ac t in c a se S = IR¡Ж, and A(S) is bounded in M by the inequali ty above (taking t = 0) in c a s e S = [ 0 ,1 ] . Then A is r e la t iv e ly co m p ac t in C°(S,M) by the A s c o l i -A r z e la T h eo rem .

Loca lly , with x = exp ° f , we have

E h(?) = E(x) = 1 <(G »f ) • 9„(f ), 8h(f)>o

* i 6 II ah{€) II о

IIah(€)||0- IIv€||0_ II v 5 M | | € l l i - c 0

u s in g G ( v ) ê 6 > 0 and ||v | | S cons t fo r v £ h * U . Thus E is locally bounding. Now

D E h( ? ) - n = < ( G » i ) - 8 h( | ) , Vn + (D20h oC)-r,>o

+ I <(D2 G=?)(r! ,9h(f)), ah(f)>o

D 2E h( ?) ( r j , r j ) = <(G°Ç)(Vr) + (D2 0h = S ) - r ) ) , Vr¡ + ( D 2 в ьЧ ) • n >q

+ 2 <(D2G .Ç ) . (n ,3h(|)), Vn + (D2V Ç) -r)>0

+ < ( G . € ) - 8 h (f) , (D26h° ! ) (n ,r | )> 0

+ i <(D2G ° ? ) 0 l , r ) , a h (?)), 8h( |)>o

ë 6 | |V + (D2 0h ° f ) • ri||q

- cons t II riling II 3h(?) | |0 II Vr)+ (D2eho | ) • n | |0

- cons t ( Il 8h ( i ) Il 0 + Il 8h(f ) Il 2 ) Il n II 2 0

6 II II2 x II II2È ~2 b ill - const II Ilco

fo r II f Hj bounded.F o r M o rse T h eo ry se e Refs [8, 9] and fo r i ts app l ica t ions on geo d e s ic s

R efs [ 5 ,7 , 9 ] .

3. VARIATION INTEGRALS

We sh a l l f i r s t c o n s id e r s p e c ia l d i f fe re n t ia l o p e r a to r s on C ” (S,M) with v a lu es in a C " v e c to r bundle E-* S X M. We have the v e c to r bundles

IA EA -SM R-11/19 129

L J(TS,TM)->- S X M of j - m u l t i l i n e a r m a p s f ro m TsS to Tx М and can, fo r any m u lt i in d e x a = (o^ , . . . ,c*r ), 1 S q>v 5 k, fo rm the v e c to r bundle

E a = L (L a i (TS, T M ) , . . . b “r(TS, TM); E ) - S X M

of r - m u l t i l i n e a r m a p s with v a lu es in E . L et к and w be g iven in te g e r s and su ppose we have a C" se c t io n A a in E a fo r ea ch m u lt i index a , with l S a y S k , a = a^ + . . . + c*r S w. Then, fo r f 6 C " (S, M), we can define

V ( “i _1 “f 1 \P(f)(s) =2^ Aa ( s , f ( s ) ) - (V 9f(s), . . . , V a f f s y

CL

to obta in a C ” s e c t io n P(f) in (id, f)’:'E -> S. We c a l l P a po lynom ia l d i f fe re n t ia l o p e r a to r of o r d e r к and w eight w on E-* S X M and denote the se t of those by PDk (E). H ere VJ9f is the co v a r ia n t d e r iv a t iv e of o r d e r j of the s e c t io n 8f in L(TS, f*TM) [3]. We c a l l P £ P D | k(S X M , B ) (E = S X M X ]R - S X M) s t ro n g ly e l l ip t ic , iff th e re is a cons tan t \ > 0, such tha t

Aa ( s ,x ) • (tok ®. ? , wk®Ç) s A0 Il u II2k II ÇII 2

fo r a - (k, k) and a l l s € S, x 6 M , u £ L ( T sS,]R), § e TXM, w here u k® f is thee le m e n t of L k(TsS, TXM) defined by u k® f (vj , . . . , vk) = u (vj).................u (v^JÇ.

We can c o n s id e r P 6 P D k (E) a s a s e c t io n in the l in e a r bundle C " (C” (S, M);:;E)-> C* (S, M), with C°°((id, f)*E) a s a f ib re o v er f, and we want to extend P to a C“ se c t io n in a l a r g e r bundle r j ( r ( S ,M ) * E ) fo r a su i ta b le m an ifo ld m o d e l Г and se c t io n fu n c to r Г-,.

T h e o r e m 6. L et P e P D f ( E ) , k > n / p , then P extends to a C°° s e c t io n in the С v e c to r bundle:

L ¿ ( L P(S, M ) * E ) - L k(S, M)

If P e PD™ (E) and к > —, к > m S pk, then P ex tends to a C " s e c t io n in the C" v e c to r bundle: ^

L k-m(Lk(S*M >S:'E ^ L k(S' M >

F o r p ro o f we r e f e r to Ref. [4] . Now we have Vm *9 £ PD™ (Lm (TS, TM)) fo r the c o v a r ia n t d e r iv a t io n of m a p s S-*M and thus by T h e o r e m 6 it is a C~ se c t io n in:

L k-m ( L k(S’ M)* Lm(TS' TM))

Vm 4 9 t i

L P(S,M)

k > *P

1 s m s к

130 ELLASSON

Then

e £: L k(S, M) -*• IR

m = l S

is a w ell defined function of c l a s s Cr , r f < le t H k(f*TM)

be the c lo s u re in Hk(f*TM) of a l l H k- s e c t io n s in f*TM having com pac t suppo r t in the in t e r io r of S, i. e. v an ish ing id e n t ic a l ly in an open neighbourhood of the b oundary 9S. Then f-> H j( f :iTM) is a C“ in te g rab le d is t r ib u t io n on H k(S,M), and we denote by H k(S,M )g the m a x im a l connected in te g r a l m anifo ld of this d i s t r ib u t io n contain ing g. Then Hk (S ,M) is a C ” subm anifo ld o f H k(S,M) w ith c h a r t s g iven by r e s t r i c t i n g the c h a r t s H k(exph) of Hk(S,M) to Hk(h*U) = Hk (h*U) П H j(h*TM ). The m a p s in H*4S,M)g have the s a m e I ^ - D i r i c h l e t bo u n d a ry va lues a s g and in p a r t i c u l a r f(s) = g(s) fo r a l l s e 3S, if f e H \ s , M ) g .

T h e o re m 7. L et g £ C “ (S,M) and P G P D kk(S X M , IR) be s t ro n g ly e l l ip tic . Then

J(f) = f P(f)s

is a C“ lo c a l ly bounding and loca l ly c o e rc iv e function on H ^S .M ) with r e s p e c t to C°(S ,M )g (or C°(S,M)).

P ro o f . In a c h a r t Hk(exph ) : Hk(h*U )-* H k(S, M), h € C " ( N , M ) , h = g in an open neighbourhood of 9S, we can w r i te the lo c a l r e p r e s e n ta t iv e of P:

Ph :H k( h * U ) - L1(S) IR), Ph (?) = P(expoÇ)

as

ph (D = T (Aa» f ) . ( v aiç , . . . , v “ri )a

1 s k , I a I 5 2k. Since P is s t ro n g ly e l l ip tic we can e s ta b l i s h a G ârd ing inequali ty of the fo rm :

IAEA-S M R-11/19 131

fo r som e c o n s tan ts > > 0, y ü, y y holding fo r a l l r) G H^(h*TM) and f G Hg(h*U) with II i ||c(j < 6 . Then fo r 6> 0 su ff ic ien tly s m a l l (X- y062 § Л/ 2 ) we get:

в ( € , 0 * 5 II ? II2 - С II? II2 for Hil l <á¿ h С

F o r а ф (к ,к) we have:

I Г “1 “r IX« = I J Ax 0 ? (v C . . . - . v 5)1

s

We choose py = 2k/av and obtain

|а |Д r - |a | /k Xa S cons t | | i | | k ||Ç||C0

S e U\\l+C , fo r llfll < e = 6 ( 0

and any e > 0. T h is follows fo r а ф (к, к) |а | = 2k, s ince then г - | а | / к = г - 2 > Оand fo r I or I < 2k us ing the e le m e n ta ry e s t im a te

_ eae b 1'9 á в e a + (1-0) e 1_6b; 0 < в < 1, e > О

which holds fo r any pos i t ive n u m b e rs a , b. Above, в = |а | /2 к < 1. Then,su m m in g up and choosing the e 's su ff ic ien tly s m a l l , we obtain

Jh( ? ) - - г N i l ! +const, Hell2 <6c°

o r J is lo c a l ly bounding with r e s p e c t to C°(S,M). Next we com pute the d e r iv a t iv e s of J h :

D Jh(? ) - r ,=J ^ ( D 2Aa o ? . ( n , v aiÇ............ , A )S a

гV «1 a ar

+ ¿ Aa o?(V "r,............... V rf))u= 1

132 ELIASSON

D2J h(Ç)- (rj.rj) = B(Ç,n) + £ Bvf¡(í) ( v V v % )v + l l < 2 k

fo r Ilf ||k S L, ||g | | 0 < 5 . Thus J is loca lly c o e rc iv e with r e s p e c t to C°(S, M)(a m o r e d e ta i led a n a ly s i s of the e s t im a te s can be found in R e f . [4] ).

V ar ia t io n in te g r a l s J a s in th e o re m 7 a r e p o ss ib ly not weakly p r o p e r , in g e n e ra l . T h is can be i l l u s t r a te d by the following exam ple : Let

Г и ь-l „2 nJ(f) = / (I V 9f II , к > — , n > 1

s

S = M = T n= IRn/ Z n and c o n s id e r the sequence fm : T n-* Tn, g iven by fm (x)=mx, m = 1, 2. . . . Then 9 fm (x):IRn->- IRn is ju s t the m u lt ip l ic a t io n by m and Vk_19f = 0 fo r к i 2 and so J ( f m ) = 0. dJ(fm) = 0. So J is bounded on the se quence f m , which obviously does not conve rge even in C°(S,M). Thus,J is not weakly p r o p e r on Hk(Tn, T n), but it could s t i l l be weakly p r o p e r on Hk(Tn, T n)g, a s a l l the m a p s f m a r e in d if fe ren t homotopy c l a s s e s . It i s , fo r exam ple , an open and in te r e s t i n g ques tion w he the r v a r ia t io n in te g ra l s like

J(f) = J (H Div 9f||2 + X II 9f ||2 )s

a r e weakly p r o p e r on H2(S,M) (see exam ple 6 h e r e and Ref. [4] ), in g e n e r a l fo r 2 > n /2 .

We sh a l l now show tha t the e n e rg y in te g r a l E¡, defined e a r l i e r is weakly p r o p e r , u s in g the following two L e m m a s p roved in Ref. [4] :

L e m m a 3. L et n < q < oo. Then th e re is a co n s tan t b, depending only on q and the R iem a n n ian m an ifo ld S, such tha t

d M (f(s), f(t)) S b d s ( s , t ) 1‘"/q ( y ' d 8f f )s

fo r a l l f e C “ (S,M) (and thus a l so in L j (S, M)) and s, t £ S , w here d M, d s denote the d is ta n c e functions in M, S.

L e m m a 4. L et 1 S p, q S oo and 0 s m S k, then th e re is a cons tan t с = c (p, q, m , k, S), such tha t fo r any E 6 VB(S) with so m e R iem ann ian m e t r i c and connection , we have

IAEA-S M R -11/19 133

pro v id ed

(with > if q = oo and then L~ is r e p la c e d by Cm ) and the n o rm s a r e defined as p re v io u s ly in s e c t io n 1.

In p a r t i c u l a r , with E = f*TM, f G L k(N, M), we obtain u n ifo rm e s t im a te s fo r n o r m s and d is ta n c e s . Thus a Cauchy sequence in the F in s l e r m anifo ld L k(S,M ), k > n / p , is a l so a Cauchy se q uence in C°(S, M), which is co m p le te , so L k(S,M) is e a s i ly se en to be a com ple te F in s l e r m anifo ld .

T h e o re m 8. The en e rg y function

E k: H k(S,M)g- R , E k(f) = I | |9f ||2k l

k > n / 2 , is w eakly p r o p e r with r e s p e c t to C°(S, M), p ro v id ed M is c o m p ac t in c a s e 3S is em pty , and s a t i s f i e s condition (C).

P ro o f . We choose q such tha t

0 < 1 - n / q S k - n /2

Then by le m m a 4, we have

II af II „ s с II af У = с (2E (f))i" HLP " k-l к

and then by le m m a 3:

d M(f(s), f(t)) 5 b c ( 2 E k(f))*ds ( s , t ) 1"n/q

fo r a l l f £ H k(S,M)s and s , t e S . Thus if E k is bounded on a su b se t A of Hk(S,M) , then A is an equicontinuous s u b s e t of C°(S,M). Now if 3S is not em pty , say s 0 G 3S, then f (s0) = g (s0) fo r a l l f in H k(S, M)g and (f(s), h(sQ)) á con s tan t , fo r f G A o r A(S) is bounded in M. T hus , in any c a s e , A(S) is a r e la t iv e ly co m p ac t s u b se t of M and then A is r e la t iv e ly com pac t in C°(S, M) by the A s c o l i -A r z e la T h e o re m . We have shown E k to be weakly p r o p e r and it is loca l ly bounding and loca l ly c o e rc iv e by th e o re m 7. Thus Ek s a t i s f i e s condition (C) by th e o re m 3.

C o ro l la ry . Let

J = J-p, P G P D 2k( S X M , IR), k > | s

s t ro n g ly e l l ip t ic and M co m p ac t , if 3S is em pty . Suppose J dom ina te s E k(J(f) ë 7 E k(f) + cons t , 7 > 0). Then J : Hk(S, M)g->■ IR is weakly p r o p e r with r e s p e c t to C°(S,M) and s a t i s f i e s condition (C).

134 ELIASSON

L et n= d im S be e i th e r 2 o r 3 and 9S be em pty . L e t M be co m p ac t and with n o n -p o s i t iv e s e c t io n a l c u rv a tu re .

Now H 2(S,M) is a com ple te C” R iem ann ian H ilb e r t m anifo ld . We cons id e r the v a r ia t io n in te g ra l :

/ 2 2 (Il A f II + Л û 3 f II ) , X > 0

S

Ш

A f(s) = Div 9f(s) = \ Vdf(s) * (еАл e. ), w ith ie-} an o r tho-n o r m a l b a s e fo r T S. Then (zis 1 - 1

J = Jp, with P G P D 2 (S X M, IR) s

s t ro n g ly e l l ip t ic , soJ : H 2(S, M) -* IR

Exam ple 6 [4].

is of c l a s s C°° and loca lly bounding and loca l ly c o e rc iv e with r e s p e c t to C°(S, M) by th e o re m 7. We have , se e Refs [ 3 ,4 ] ,

V Div 9f • v = ^ V 9f • (v, e . , e. )Í

= (V 9f ( e . , v, e. ) + RM » f (9f • v , 9f • e ^ 9f- e. - 9f- Rs(v, e. )e. ) i

= (p iv V 8 f + ^ R Mof(9f- , 9. f)9.f - 9f R icg) - v i

with R M, R g the c u r v a tu re t e n s o r s of M, S and R ic s the R icc i t e n s o r of S and 9jf = 9f • e , . T hen

2 J(f) = - <VDiv 9f, 9f> + X II 9f ||2 'о 0

= - <D iv V9f, 9f> - < Y r of(9f , 9, f) 9 ,f, 9f>o Z_) m 1 1 о

i

+ < 9 f R ie . , 9f> + X II 9f II 2S ' 0 1 о

5 II V 9f II - K J II 9f ||4 + (X + XQ) II 9f ||2s

with XQ the m in im u m of a l l e igenva lues of R ics at a l l tangent sp a c e s of S and К the m a x im u m of a l l s e c t io n a l c u r v a tu r e s on M. Now, we have a s s u m e d K s 0 and then choos ing X ê 1 - X0 , we obtain:

IAEA -SM R -11/19 135

j(f) S i II 3f ||2 = E2(f)

Thus J is w eakly p r o p e r with r e s p e c t to C°(S,M) by the l a s t c o r o l l a r y and s a t i s f i e s condition (C). The c r i t i c a l po in ts of J a r e m a p s of c l a s s C" by th e o re m 16 [4] and if f E C “ (S,M), then

dJ(f) • 5 = <Af, A ? + R M° f ( C , 9 . f ) a f > 0 + A<9f, V?>o

= <ЛД f + RM ° f (Дf, 9. f) 9 jf - А Д f , O o

T hus dJ(f) = 0 im p l ie s , t a k i n g ! = - A f :

0 = H V Д f ||2 - < R Mo f(A f ,9 . f )9 . f , Af>o + X \\A{ ||¡

and s in c e a l l th r e e t e r m s a r e n o n -n eg a t iv e , we m u s t have A f = 0, o r f : S-* M is a h a rm o n ic m ap . By th e o re m 2, J ta k es i ts m in im u m in e v e ry com ponent of H2(S,M) in a h a rm o n ic m ap and we have the lo w er bounds fo r the n u m b e r of h a rm o n ic m a p s g iven th e re . T h is im p ro v e s the r e s u l t s of E e l l s and S am pson [2] s l igh t ly in c a s e n= 2, 3.

R E F E R E N C E S

[1] EELLS, J., Jr., On geometry of function spaces, Symp. In t.T opol.A lgebr. (Mexico 1956) (1958) 303.[2] EELLS, J., Jr., SAMPSON, J .H ., Harmonic mappings of Riemannian manifolds. Amer. J. M ath. 86

(1964) 109.[3] ELIASSON, H .I . , On the geometry of manifolds of maps, J.D iff.G eom . 1 (1967) 169.[4] ELIASSON, H .I . , Variation Integrals in Fibre Bundles, Proc.Sym p. Pure Math. 16, A m er.M ath .Soc.,

(1970) 67.[5] ELIASSON, H .I ., Moers Theory for closed curves, S ym p.Inf.D im .T opol.A nn. of M ath.Studies 69

(1972) 63.[6] ELIASSON, H .I ., Condition (C) and geodesics on Sobolev manifolds, B ull.A m er.M ath.Soc. 77 6

(1971) 1002. ”[7] GROVE, K ., Condition (C) for the energy integral on certain path-spaces and applications to the theory

of geodesics, Preprint series 1971/72, N o.4, Aarhus Universitet, Denmark.[8] MEYER, W ., Kritische M annigfaltigkeiten in Hilbert — M annigfaltigkeiten, M ath.Ann. 170 (1970) .[9] PALAIS, R .S ., Morse theory on Hilbert manifolds, Topology 2 (1963) 299.

[10] PALAIS, R .S ., Foundations of Global N on-linear Analysis, Benjamin, New York (1968).[11] PALAIS, R .S ., Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966) 115.[12] PALAIS, R .S ., C ritical point theory and the m inim ax princip le . Proc.Sym p.Pure M ath. 15, Amer.

M ath .Soc., 185.[13] PALAIS, R .S ., SMALE, S ., A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964) 165.[14] SCHWARTZ, J .T ., Generalizing the Lusternik-Schnirelman theory of C ritical points, Comm. Pure

A ppl.M ath. 17 (i964 ) 307.[15] SMALE, S ., Morse theory and a non-linear generalization of the D irichlet problem, A nn.M ath. (2)

80 (1964) 382.[16] UHLENBECK, K .K ., The Calculus of Variations and Global Analysis, Dissertation, Brandéis University,

W altham, Mass. (1968).

IA EA -SM R-11/20

ELEMENTARY SURVEY OF PSEUDO-DIFFERENTIAL OPERATORS AND THE WAVE-FRONT SET OF A DISTRIBUTION

R.J. ELLIOTT Mathematics Institute,University of W arwick,Coventry, Warks,United Kingdom

Abstract

ELEMENTARY SURVEY OF PSEUDO-DIFFERENTIAL OPERATORS AND THE WAVE-FRONT SET OF A DISTRIBUTION.

Following Hormander and Sato, the author, successively, deals with the theory of operators with constant coefficients, of operators with variable coefficients and pseudo-differential operators and with the theory of the w ave-front set.

The r e s u l t s and id e as p r e s e n te d in th is p a p e r a r e due to H o rm a n d e r and Sato; the t r e a tm e n t follows H o rm a n d e r [2, 3].

1. OPERATORS WITH CONSTANT COEFFICIENTS

We sh a l l u se the s ta n d a rd m u l t i - in d e x notation: a = (ai , an) is an - tu p le of non -n eg a t iv e in te g e r s .

If x £ R m , x = x ^ xg2 ... x “n , and if D = ^ - i , . . . , - i ,

D “ = ( - i ) ,a | ( ^ 7 . •••, * w h e re M = “ i + “ 2 ••• + an •

A l in e a r p a r t i a l d i f fe re n t ia l o p e r a to r with cons tan t coeff ic ien ts can then be w r i t te n

P(D) = Y a « D“|a|^m

w h e re a a a r e com plex n u m b e rs . If f e C o (Rn ) = ¿Z>(Rn), i ts F o u r i e r t r a n s f o r m is defined as

л - n /2 Г -i'C x, 6/f (?) = (27t) J e f(x) dx

Rn

137

138 ELLIOTT

w h e re dx is L ebesgue m e a s u r e . It i s e a sy to check , using in te g ra t io n by p a r t s , tha t

- ^ - ( ? ) = ( 1 Л >

C o n s id e r the equation

P(D) U = f (1.2)

f o r f e C o (Rn). If th e r e is a so lu t ion u of Eq.(1.2) with a w e l l -de f ined F o u r i e r t r a n s f o r m u(?) then , f ro m a su i tab le ex tens ion of E q . ( l . l ) :

P (? )û ( ? ) = f (?)

So by F o u r i e r ' s in v e rs io n fo rm u la , we have

u(x) = (2тг)"п/2 [ е ‘ <х’ £> f (? ) /P (? )d ? (1.3)

H ow ever , P m a y have z e r o s on Rn ; so , to obta in a so lu t ion , it m ay be n e c e s s a r y to d e fo rm the path of in te g ra t io n f ro m Rn into C n . To show th a t r e la t io n (1.3) can a lw ays be in te r p re te d we f i r s t p rove two le m m a s .

L e m m a 1 .1 . Suppose $ e C c ( C n) and Ф(е10 ?) = Ф ( ? ) , 6 e R (*)

and / ® (?)dX (?) = l CH<)

w h e re dX(?) is L ebesgue m e a s u r e on Cn . Then, fo r any ana ly tic function F on Cn ,

J F(Ç) Ф(?) dX (?) = F(0) (1.4)

P r o o f . By C auchy1 s fo rm u la

2тг

f(? e i0)d6 = 2ttF(0)

So2тг

J Ф(?) F(C e i9 ) dedX (?) = 2n F(0)

Cn 0

IA EA-SM R-11/20 139

but

J Ф(f) F (f e i0 ) dX (?) = J 4 > (? )F ( | )d X ( | )

C n C n

and so the r e s u l t is p roved .Denote by Pol(m ) the com plex v e c to r sp a c e of po lynom ia ls in

n - v a r i a b l e s of d e g re e § m . W rite P o l° (m ) fo r the sp a ce with the o r ig in re m o v e d . F o r Q e P o l ( m ) , w r i te Q(“>(Ç) = (iD)aQ fo r any m u l t i - in d e x a ; then

| | q || = £ |q (“ ! (0)|

|aj£ m

def ines a n o r m on Pol(m ).

L e m m a 1 .2 . If Г2 is a neighbourhood of O e C n th e re is a C " m ap

Ф: P o l0 (m)-> С0”(П)

hom ogeneous of d e g re e z e ro , such tha t the ra n g e functions s a t i s fy condit ions (*) and (**) and fo r som e c o n s tan t K:

| | q || S К |Q(?) | , Q e P o l 0 (m) (1.5)

and I e supp. $(Q).

P r o o f . F o r a fixed Q e P o l ° ( m ) , th e r e is a 0 e R n such tha t

Q(z0) f 0 fo r |z | = 1

F o r th is Q choose а Ф w ith su p p o r t n e a r the c i r c le zв. An a p p r o p r ia te ly m odif ied Ф s a t i s f ie s condition (1.5) and conditions (*), (**). A lso th is Ф can be u sed fo r a l l n e a r b y p o lynom ia ls . The s e t of functions sa t is fy ingcondit ions (*), (**) is convex so the c o n s t ru c t io n of Ф can be f in ished bym e a n s of a p a r t i t io n of un ity on the s e t {Q: | |Q 11 =1}.

F in a l ly , we can show how to i n t e r p r e t r e la t io n ( 1 . 3 ) .

T h e o r e m 1 . 3 . T h e re e x i s t s a continuous m ap E: P o l°(m ) -* &)' ( R n ) such th a t P(D)E(P) = 6, the D ira c 6 -func tion , fo r e v e ry P e P o l ° ( m ) .

P r o o f . C o n s id e r the e x p r e s s io n

(Ef)(x) = (2ж ) ' П/2 / d? f e 1<x’ £+S>(f(f + + Ç)) Ф ( P £ , 5) dX(f)

Rn C n ( 1 . 6 )

where P j i s the polynomial

Ç - P (Ç + Ç)

140 ELLIOTT

Since one d e r iv a t iv e of P is n o n - z e ro the function

P(€) = ^ | p (0° (C ) | , ? e R n

|a|^m

has a p os i t ive lo w e r bound. T hus , by le m m a 1.2, P is bounded away from z e r o in the su p p o r t of the f i r s t in teg ran d and so e x p r e s s io n (1.6) is well defined fo r feCci°(Rn). By le m m a (1.1), if we d if fe re n t ia te u n d e r the i n te g r a l we have

P(D)(Ef) = f

and so we have so lved Eq.(1.2) fo r f e C Ô ( R n).The m ap f -» Ef co m m u te s with t r a n s l a t i o n s so th e re is a d is t r ib u t io n

which we a l so denote by E such tha t E f = E*f. As (P(D)E)* f = f fo r a l l f e C 0"°(Rn) we have P(D)E = 6, the D ira c 6 -function.

If f e á ”' (Rn), the sp a ce of d is t r ib u t io n s with com pac t su p p o r t , the d i s t r ib u t io n u = E*f i s w e l l defined and s a t i s f i e s Eq.(1 .2) .

D efin it ion 1 .4 . The d is t r ib u t io n E is ca lled a fundam enta l so lu tion fo r the o p e r a to r P .

2. OPERATORS WITH VARIABLE CO EFFICIENTS AND P S E U D O -D IF F E R E N T IA L OPERATORS

In the c a se of co n s tan t coe ff ic ien ts above, the r e s u l t s depended e s s e n t i a l l y on the F o u r i e r t r a n s f o r m , which in te rc h a n g e s (see E q . ( l . l ) ) d if fe ren t ia t io n and m u lt ip l ica t io n . H ow ever , d i f fe ren t ia t io n and m u l t i p l ic a t io n do not th e m s e lv e s com m ute so fo r o p e r a to r s with v a r ia b le co e f f ic ien ts one ap p ro x im a te s a fundam en ta l so lu t ion by co n s t ru c t in g a p a r a m e t r ix .

Using the above no ta tion , c o n s id e r a d i f fe re n t ia l o p e r a to r P with v a r ia b le coe ff ic ien ts

P ( x , D ) = ^ T a„(x) D“ (2.1)

a ^ m

in an open s e t X c R n . We a s s u m e a a ( x ) e C w(X) and suppose

P (x , D) = Pm (x, D) + Pm-! (x, D) + ...

i s a dec o m p o s i t io n of P in a sum of hom ogeneous t e r m s Pj of d e g re e j.

D efin it ion 2.1

P is sa id to be e l l ip t ic if fo r x*X and о f |«Rn

Pm (x,«) = £ a « (x )Ç “ f 0

|ot[ = m

( 2 . 2 )

IAEA-SM R-11/20 141

If P is an e l l ip t ic o p e r a to r with c o n s tan t coe ff ic ien ts we have fo r som e c o n s tan t K:

I? Г s К I P ( f ) | , If |> К

f o r f GRn o r , indeed , f in so m e n a r r o w cone in Cn contain ing R n . Suppose x e C " (Rn ) is 0 fo r | f I < К and 1 fo r l a rg e | f . | . A p a r t f ro m in te g ra t io n o v e r a com pac t s e t , w hich c o n t r ib u te s an e n t i r e ana ly tic t e r m , the fundam e n ta l so lu t ion c o n s t ru c te d in th e o re m 1.3 is s im p ly

Ef(x) = (2тг)'п/2 J e i<x’ E> f ( f ) x ( f ) /P (? ) d f (2.3)

Rn

A gain d if fe ren t ia t in g u n d e r the in te g r a l g ives a d i s t r ib u t io n E such th a t E f = E*f and

P(D)E = 6 + R (2.4)A

H e re R = x - 1 so th a t R e C ” . T h e r e f o r e , E is ' a lm o s t ' an in v e r s e fo r P and is ca lled a ' p a r a m e t r ix ' .

Now,

E = (2ir ) ~n/2 J е 1<х' ё> X ( f ) / P ( f ) d f

Rn

SO

( -x)“ E = (2тг)-п/2 J e 1^ - ^ D “ ( x ( f ) / P ( f ) ) d f

The in te g ran d d e c r e a s e s r a p id ly at infin ity , and so as th is iden ti ty is t r u e fo r a l l a we se e tha t E is C“ away f ro m the o r ig in .

D efin it ion 2.2

F o r v e g 1 (Rn) s ing . supp. v is the s m a l l e s t c losed s e t such th a t v i s C " in the c o m p lem en t , i .e .

s ing . supp. v = n { x : <p(x) = 0} (2.5)

the in te r s e c t io n being o v e r a l l <peC°°(Rn ) with <pveC°°(Rn). Note th a t if v e c f (Rn )

v = E *P(D )v - R*v

O utside s ing . supp. P(D)v, E * P ( D ) v e C " a s E e C ” ou ts ide the o r ig in and R * v e C " . T h e r e f o r e , if we have a p a r a m e t r ix fo r an o p e r a to r P , we can say: s ing . supp. v = s ing . supp. Pv.

142 ELLIOTT

R etu rn ing to e l l ip t ic o p e r a to r s on X with v a r ia b le coe f f ic ien ts , it i s , t h e r e f o r e , of i n t e r e s t to t r y and c o n s t ru c t a p a r a m e t r ix . The c l a s s i c a l m ethod of E .E . Levi w as to ' f r e e z e ' the coe f f ic ien ts a t x o 6 X and to t r y and find a p a r a m e t r ix a s a p e r tu rb a t io n of the p a r a m e t r ix E 0 of P (x 0,D ) . T hus , E 0 has the fo rm

E 0 f(x) = (2 7Г ) "n/2 f e i<x' £> dg

Rn

This p a r a m e t r ix is a b e t t e r ap p ro x im atio n to a p a r a m e t r ix of P (x , D) at x 0 than e l s e w h e re , so one hopes a b e t t e r ap p ro x im a tio n is obta ined if P (x 0, ?) is r e p la c e d by P (x , | ) . Note th a t P (x , f ) " 1 = Pm (x, f)"1 + ... w h e re the dots ind ica te hom ogeneous t e r m s of o r d e r s - m - 1 , - m - 2 , ... r e s p e c t iv e ly . One i s , t h e r e f o r e , led to c o n s id e r o p e r a to r s of the fo rm

E„f(x) = (2*)-n/2 J ei<XlE> * ( x , € ) f ( S ) d f (2.6)

Rn

w h e r e , a s Ç -► °o, <p looks like an a s y m p to t ic s e r i e s of hom ogeneous func tions of ? , of d e c r e a s in g d e g r e e s of hom ogeneity .

In fac t , as in e .g . H o rm a n d e r [1], i t is m o r e convenient to suppose tha t </>eC” (XX Rn ) and fo r so m e r e a l m and a l l m u l t i - in d ic e s a and |3

| d “ d ®<p (x , ? ) I S Ca , e, K ( l + |C |)m‘ N (2.7)

fo r x in ea ch co m pac t s u b s e t К of X. The s e t of a l l functions sa t is fy ing* (2.7) is denoted by Sm(X X Rn ).

D efin it ion 2.3

An o p e r a to r of the fo rm (2.6) with (peSm(X X R n) i s ca lled a p se u d o d i f fe re n t ia l o p e r a to r of o r d e r m with sy m b o l <p.

It can be checked th a t the m ap E^, is continuous f ro m C(j°(X) to C “ (X) and th a t i t can be extended to a continuous m ap f ro m S' (X) to 3)' (X). In f ac t , E y is a d i s t r ib u t io n on XX X and the d iagonal in XX X contains a l l the s in g u la r i t i e s of E ?. F r o m th is one h as :

s ing . supp. Ev С s ing . supp. v (2.8)

fo r a l l v e S' (X).T h e se p r o p e r t i e s a r e a l l e s ta b l ish e d in H o rm a n d e r [1].To find a p a r a m e t r ix fo r the e l l ip tic o p e r a to r , we m u s t find a <p

such tha t

P (x, D + i ) <p (x, §) - l e S " “ =П Smm

We do th is by choosing a <p with an a sy m p to t ic expansion cpo +<Pi + ••• w h e re is hom ogeneous in С of d e g re e - m - j and:

P (x, D + Ç) (<p0 + <Pj + ... + <pj) - l e S J 1 , j = 0, 1 ...

IAEA-SM R-11 /20 143

<Po = Рш

and the conditions can then be s a t is f ie d r e c u r s iv e ly . T h is m ethod is ju s t a n o th e r way of c a r ry in g out the c l a s s i c a l i t e r a t iv e solv ing of in te g r a l equa tions w hich o c c u r s in the L ev i m ethod .

Thus p s e u d o -d i f f e re n t i a l o p e r a to r s g ive a convenien t f ra m e w o rk fo r c o n s t ru c t in g p a r a m e t r i c e s fo r e l l ip t ic o p e r a to r s . H ow ever , they a l so fo rm a n a tu r a l ex tens ion of the c la s s of d i f fe re n t ia l o p e r a to r s : a d i f fe re n t ia l o p e r a to r P (x ,D ) is of the fo rm (2.6) with <p(x,f) = P ( x , f ) . In fac t , p s e u d o d i f fe re n t ia l o p e r a to r s fo rm an a lg e b ra which i s in v a r ia n t u n d e r taking ad jo in ts and change of v a r ia b le s .

D efin it ion 2.4

A p s e u d o -d i f f e re n t i a l o p e r a to r P of o r d e r m and sy m b o l p(x, f ) is c h a r a c t e r i s t i c at ( x J ) e X X (Rn\ 0 ) if

l im |p(x, tf ) | t ' m = 0 t-» + “

The c h a r a c t e r i s t i c po in ts of P fo rm a c losed cone in X X (Rn \ 0 ) .L et us w r i te c h a r P fo r the s e t of c h a r a c t e r i s t i c points of P . If no c h a r a c t e r i s t i c e x i s t s , we sa y th a t P is e l l ip t ic and the above co n s t ru c t io n aga in then shows tha t we can find an o p e r a to r Q of o r d e r -m so tha t Q P - 1 = Rl5 and PQ - 1 = Rg both have C ” k e r n e l s . Thus aga in fo r e l l ip t ic p s e u d o -d i f f e re n t i a l o p e r a to r s , we have:

s ing . supp. v = s ing . supp . P v; v e á ^ (X)

3. THE W AVE-FRO N T SET

Using the above id e as we w ish to c lo se with a few r e m a r k s concern ing the wave f ro n t s e t of a d is t r ib u t io n . T h is in te r e s t in g concep t was f i r s t in tro d u c ed by Sato [4], in connect ion with h is hyperfunc t ions .

In defin ition 2.2 we defined the s in g u la r su p p o r t of a d is t r ib u t io n v e ^ 1 (X) as:

s ing . supp. v = n {x; <p(x) = 0}

the in te r s e c t io n being o v e r a l l <pe.C “ (X) with <pveC“ (X).

D efin it ion 3.1

The w a v e - f ro n t s e t of a d is t r ib u t io n ve¿2>' (X) is

WF(v) = П c h a r A

the in te r s e c t io n being o v e r a l l p s e u d o -d i f f e re n t i a l o p e r a to r s A such tha t A u e C " ( X ) .

WF(v) is a c lo sed cone in X X (Rn \ 0 ) .

F o r j = 0, therefore

144 ELLIOTT

The p ro je c t io n of WF(v) on X equa ls s ing . supp. v.

P r o o f : C le a r ly , the p ro je c t io n of WF(v) on X is contained in s ing . supp. v.If x is not in the p ro je c t io n of WF(v), then we can find f in ite ly m any p se u d o d if fe re n t ia l o p e r a to r s A¡ with A j v e C , A rA j w ell defined and {x}X (Rn\ 0 ) n (O c h a r Aj) = j). Then if A = ¿ / A*Aj we have A v e c " and A is e l l ip t ic at x so v is C“ th e re .

In the defin ition of the w a v e - f ro n t s e t , we s e e tha t i t is enough to c o n s id e r o p e r a to r s of o r d e r 0. Indeed, i t is enough to c o n s id e r o p e r a to r s of the fo rm b(D)a(x), w h e re b ( i ) is a hom ogeneous function of d e g re e 0 fo r l a rg e IС | . As w as p roved in Ref. [3], th is leads to an equ iva len t def in i t ion , which is quite i l lum ina ting :

T h e o re m 3.3

(x0 , ! 0 ) ^ WF(v) if and only if fo r so m e ne ighbourhood of x 0 one can find u e S' (X) equa l to v in th is ne ighbourhood and v(Ç) = 0( |ç | ' N ) fo r e v e ry N in a conic ne ighbourhood of §o independent of N.

F in a l ly the notion of the w a v e -f ro n t s e t ind ica tes when we m ay define the p ro d u c t of two d is t r ib u t io n s Uj, (Rn). Suppose x e C j f R 11),¡X dx = 1 and put X £ (x) = e"n x ( x /e ) . We should l ike to define u i u 2 a s the l im i t a s e -> 0 of (и 1* Х €) ( и 2* Х £ )• In g e n e ra l , the l im i t does not ex is t . H ow ever , if

WF(U1) + WF(U2) = {(x, I j + Ç2 ); (x ,C i)e W F (U i)} c X X (Rn \0 )

the l im i t does ex is t . F u r t h e r the l im i t is then independent of x*F in a l ly , le t us r e p e a t th a t we have only ske tched ideas and r e s u l t s

due to H o rm a n d e r and Sato , the fu ll d i s c u s s io n of w hich can be found in the r e f e r e n c e s .

R E F E R E N C E S

[1 ] HORMANDER, L ., "Pseudo-differential operators and hypoelliptic equations ", Am. Math. Soc.Symp. Pure M ath. 10 (1966); Singular integral operators, 138.

[2 ] HORMANDER, L ., "On the existence and regularity of solutions of linear pseudo-differential equations", L’Enseignement Math. (1971)99.

[3] HORMANDER, L ., "Fourier integral operators. I", Acta Math. 127 (1971) 79.[4 ] SATO, М ., "Regularity of hyperfunction solutions of partial differential equations", Actes Congrès

Intern. M ath ., N ice (1970).

Theorem 3.2

IA EA -SM R-11/21

BOUNDARY VALUE PROBLEMS FOR NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

R.J. ELLIOTT Mathematics Institute,University o f Warwick,Coventry, Warks,United Kingdom

Abstract

B O U N D A R Y VALUE PROBLEMS FOR NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS.The paper deals with differential games and the differential equations by which they are governed;

in a special section the relevant boundary value problems are discussed. In the last section, so-called "games of survival” are studied.

INTRODUCTION

The r e s u l t s below a r e r e c e n t w o rk of Kalton and the au th o r [4, 6], g e n e ra l iz in g ideas of F le m in g [ 7 ] .

Suppose t e [ 0, 1], x e Rm, and Y i s a co m p ac t m e t r i c space . C o n s id e r a dy n am ica l s y s t e m x(t) = f(t, x, y),x(0) = x0)1 w here f i s continuous andL ip sc h i tz in x, and a quan ti ty P = g(x(l)) + / h(t, x, y)dt. H ere h : Rm+1 x Y-*Ris continuous and g is a r e a l -v a lu e d function.

O p tim a l co n tro l p ro b le m s s tudy s i tu a t io n s of th is kind, and the ob ject is to d e te rm in e a function y(t) so tha t P (the "cos t" ) i s a s s m a l l a s p o ss ib le . If Z is an o th e r c o m p ac t m e t r i c space , an ex tens ion of th is s i tu a t io n i s to c o n s id e r a d y n am ica l s y s t e m x(t) = f(t, x, y, z), and r e a l -v a lu e d "pay-off"P(y, z) = g(x(t)) + J h(t, x, y, z)dt. We suppose tha t th e re a r e two c o n t ro l le r s o r p la y e r s : who c o n t ro ls y G Y and who is try in g to m ake P a s la rg e a sp o ss ib le , and J2 who c o n t ro ls z e z and who is t ry in g to -m ake P a s s m a l la s po ss ib le . Such a s i tu a t io n i s ca l le d a "d i f f e re n t ia l gam e" .

D IF FE R E N T IA L GAMES

D if fe re n t ia l g am es w ere f i r s t s tud ied in the 1950 's in r a th e r a fo rm a l w ay by I s a a c s , though h is r e s u l t s w e re not p ub lished un til 1965 [ 10] . The p ro b le m when d e s c r ib in g a r ig o ro u s m a th e m a t ic a l m ode l fo r a d i f fe ren t ia l gam e i s the in fo rm a tio n p a t te rn ; idea lly , a t any in te rm e d ia te t im e each p la y e r should know w hat the o th e r h a s done up to tha t t im e but should not know w hat he w ill do in the fu tu re . To m ake th is idea p r e c i s e F le m in g [ 8] ap p ro x im a te d the s i tua t ion by r e p la c in g the d i f fe re n t ia l equa tions by d if fe ren ce equa tions and co n s id e r in g a p a r t i t io n of the t im e in te rv a l into 2n p ie c e s Ij , j = 1, . . . 2n. In the u p p e r gam e co r re sp o n d in g to th is p a r t i t io n J 2 p lays f i r s t on each in te r v a l and in the lo w e r gam e, J2 p lays second . T h e se g am es have va lu es W* and W" and F le m in g show s th a t a s n -> 00 th e se co nve rge to quan t i t ie s W + and W \

145

146 ELLIOTT

In h is w o rk [ 9] F r ie d m a n does not ap p ro x im a te by d if fe ren ce equations but he does c o n s id e r a p a r t i t io n of [ 0, 1] and co r re sp o n d in g upper and lo w er gam es w h ere J 2 p lays f i r s t o r second on each Ij. In the l im it , F r i e d m a n ob ta in s u p p e r and lo w er v a lu es V+ and V ' and he is ab le to show V + = V" if the opposing co n tro l v a r i a b le s a r e s e p a r a te d in f and h, i. e. if

f(t, x, y, z) = fj(t, x, y) + f2(t, x, z)

h(t, x, y, z) = h jjt , x, y) + h 2(t, x, z)

W r i te fo r the sp a c e of m e a s u r a b le co n tro l functions f ro m [0, 1] toY and ..< 2 f ° r the s i m i l a r sp a ce with va lues in Z. A m ap a -^l ca l le d a s t r a t e g y fo r if i t i s n on -an t ic ipa t ing , i. e. if, fo r any T S 1, Z j ( t ) = z2(t)a . e. 0 S t É T, then ( a z j (t) = (a z2) (t) a. e. 0 É t â T.

A s t r a t e g y a h a s a value

u(a) = inf P (a z, z) zGjc,

and th e re is then an u p p e r value U = sup u(a).S t ra te g ie s (3 :. л ^ - » a r e defined s i m i l a r l y and v(|3) = sup P(y, /Зу)

y 6 ^ iV = inf v(/3)

8

In Ref. [ 2] Kalton and the au th o r show tha t if the I s a a c s m in im ax condition holds, i . e . fo r a l l t G [ 0, 1 ], x , p £ R m

m in m ax (pf + h) = m ax m in (pf + h) z y y z

then a l l the above v a lu e s a r e equal: W ' = V ' = V = U = V + = W +. This is a s t r o n g e r r e s u l t than tha t of Ref. [ 9 ] .

D IF F E R E N T IA L EQUATIONS

Now c o n s id e r a gam e s ta r t in g f ro m so m e in te r m e d ia te t im e and posit ion (t, x) G R m+1. Then the gam e s t a r t in g a t th is pos it ion w ill c e r ta in ly have an (upper) va lue U(t, x) w hich fo r a s m a l l t im e 6 w ill be r e la te d to the value U(t + 6, x(t + &)) by a " d y n a m ic -p r o g r a m m in g " - ty p e identity:

t + 6

U(t, x) = m in m ax J' h dt + U(t + 6, x(t + 6)t

If U w e re d if fe ren t ia b le then, fo rm a l ly , we should obta in th a t U s a t i s f ie s the " I s a a c s -B e l lm a n " equation:

LU = + m in m ax (VUf + h) = 0dt z ' y

with the boundary condition

U(l, x) = g (x)

IA EA -SM R-11/21 147

T h is equation i s h igh ly d e g e n e ra te and th e re a r e no r e s u l t s g u a ran tee ing ex is te n c e o r un iqueness of so lu t ions . H ow ever, if "white n o ise" is in tro d u c ed into the d y n am ics then the expec ta t ion Ux of the u pper value can be shown to s a t i s f y the n o n - l in e a r p a ra b o l ic equation

X2 V2 + 3U ^ /3 t + m in m ax (VU^f + h) = 0 z y

fo r which so lu t ions a r e known to ex is t . A s X ->■ 0, can be shown to conve rge to U. U nder L ip sc h i tz conditions on f and h, U is L ip sc h i tz and so i s a g e n e ra l iz e d so lu tion of the I s a a c s - B e l lm a n equation, in the s e n se tha t i t s a t i s f i e s the equation a lm o s t ev e ry w h e re .

BOUNDARY VALUE PROBLEM S

In h is p a p e r [ 7] F le m in g changes h is point of view. He c o n s id e r s a n o n - l in e a r equation

L*u = # + G(t,x ,Vu) = 0, t e [ 0 , 1 ] , x e Rm dt

to g e th e r with a t e rm in a l boundary condition u ( l , x) = g(x), and he then c o n s t ru c ts a d if fe re n t ia l gam e with dynam ics :

and pay -o ff

w here

H e re

x(t) 1 + 1 у 12 У

1P = g ( x (1 ) ) + J h dt

0

1 + I у I2 y

У e Y = I y G R m : I y I S <*]■

z e z = I z e R m : I z f s |3 j-

If G i s L ip sc h i tz i t i s e a s i l y se e n tha t fo r su i tab le a and j3, fo r p e RnI p I S a and any t, x:

m ax m in (pf + h) = G(t, x, p)У z

148 ELLIOTT

T h e r e f o r e , in th is c a se the lo w e r -v a lu e function i s a g en e ra l iz ed so lu t ion of the r e la t e d I s a a c s - B e l lm a n equation, which i s ju s t the given equation.

F u r t h e r m o r e , a s an y d if fe re n t ia l gam e so lu tion i s the l im i t of the unique so lu t ions of the r e la te d p a ra b o l ic equation the d i f fe ren t ia l gam e so lu t ion i s unique.

GAMES OF SURVIVAL

So fa r , we have c o n s id e re d f ix e d - t im e d i f fe re n t ia l gam es . C o n s id e r a s e t F с R m+'L such th a t F з [ T, to) x R m f o r som e T . The d if fe re n t ia l gam e s t a r t s a t ( t 0, x 0) and ends the f i r s t t im e t F tha t the t r a j e c t o r y e n t e r s F , the " t e r m in a l s e t" . Suppose the pay -o ff is then

'f

P = g(x(tp )) + J h(t, x, y, z) dt

'oIf g = 0 and h = 1, P i s ju s t the t im e to the " c a p tu re t im e" t F and so the gam e i s one of " p u r s u i t - e v a s i o n " . F o r g e n e ra l g and h, such g am es a r e ca l le d " g a m e s of su rv iv a l " , and they w e re s tud ied by Kalton and the au th o r in Ref. [ 4 ] . The I s a a c s - B e l lm a n equation fo r a game of s u rv iv a l i s m o r e c o m p lic a te d b ec a u s e the boundary condition is now

U(t, x) = g(t, x) fo r (t, x) e 3F

K alton and the au th o r show tha t if th e re a r e C 1 - functions and 02 such tha t L S 0 S L 02 and = в2 - g on 3F then U f o r the s u rv iv a l game is L ip sc h i tz continuous.

In Ref. [ 6] we apply these ideas to n o n - l in e a r boundary p ro b le m s of the form :

L*u = | ^ + G(t, x, Vu) = 0

with u = g on 3F .If th e re a r e C 1 functions O jand в2 such tha t

L*e¿ S 0 S L*02

and = в2 = g on 3F, we c o n s t ru c t , a s above, a r e la t e d d i f fe re n t ia l game, which i s now a gam e of su rv iv a l . By the r e s u l t s of R e fs [4 , 5], the value of th is d i f fe re n t ia l gam e is a so lu t ion of the boundary value p ro b lem , and aga in i t can be shown tha t the " d i f f e r e n t ia l -g a m e so lu tion" i s independent of how a d i f fe re n t ia l gam e i s a s s o c ia te d with the equation.

R E F E R E N C E S

[1] ELLIOTT, R .J ., KALTON, N .J ., Values in differential games, Bull. Amer. Math. Soc. 78 (1972) 427.[2] ELLIOTT, R .J ., KALTON, N .J ., The existence of value in differential games, Amer. Math. Soc.

Memoir 126, Providence R. I. (1972).

IA EA -SM R-11/21 149

[3] ELLIOTT, R .J ., KALTON, N .J ., The existence of value in d ifferential gam es of pursuit and evasion,to appear in J. Diff. Equations.

[4] ELLIOTT, R .J ., KALTON, N .J ., Cauchy problems and gam es of survival, to appear.[5] ELLIOTT, R .J ., KALTON, N .J ., Upper values and stochastic games, to appear.[6] ELLIOTT, R .J . , KALTON, N .J ., Boundary value problems for non-linear partia l d ifferential operators,

to appear.[7] FLEMING, W .H ., The Cauchy problem for degenerate parabolic equations, J. M ath. Mech. 13

(1964) 987.[8] FLEMING, W .H ., The convergence problem for differential gam es-II, Ann. Math. Study 52 (1969) 195.[9] FRIEDMAN, A ., D ifferential Games, W iley-Interscience, New York-London (1971).

[10] ISAACS, R. , D ifferential Games, John Wiley, New York-London (1965).

IAEA -SM R -11/22

GAUSSIAN MEASURES ON BANACH SPACES AND MANIFOLDS

K.D. ELWORTHY Department of Mathematics,University o f Warwick,Coventry, Warks,United Kingdom

Abstract

GAUSSIAN MEASURES ON BANACH SPACES AND MANIFOLDS.The paper presents some facts concerning probability Borel measures on infinite-dim ensional Banach

spaces and manifolds. The subjects discussed are cylinder set measures, weak distributions, radonification; abstract Wiener spaces; cylinder set measures versus abstract Wiener space; transformation o f integral formula; abstract Wiener manifolds; divergence theorems and theL ap lacian .and Brownian motion on a compact m anifold.

This con tr ibu t ion is b ased on tha t by E e l ls and the au thor in Ref. [6]; how ever, h e r e the e m p h a s is w ill be m o r e on the a b s t r a c t theory . Since the su b je c t is r a t h e r novel to m o s t m a th e m a t ic ia n s it s e e m s w orth pointing out th a t m e a s u r e s on in f in i te -d im e n s io n a l sp a c e s a r e not u n n a tu ra l ob jec ts d r e a m e d up by p u re m a th e m a t ic ia n s but the b a s ic s tu ff of p robab il i ty th e o ry , of im p o r ta n c e in en g in ee r in g (e. g. s to c h a s t ic co n t ro l theo ry ) , and of in c re a s in g in t e r e s t in m a th e m a t ic a l phys ic s ; r e c e n t ly , i ts in te ra c t io n with functional an a ly s is has p ro v ed e x t re m e ly r i c h [13, 14].

A. GENERALITIES

We sh a l l be dea ling with m e a s u r e s defined on topo log ica l sp a ce s , u su a l ly s e p a r a b le and m e t r i s a b l e . O u r m e a s u r e s w ill be a s s u m e d to be countably additive , p os i t ive m e a s u r e s defined on the B o re l field of the space ; an in t roduc t ion can be found in Ref. [3]. If (X,ju) is a sp a ce with m e a s u r e /к and Y is a topo log ica l sp a c e a m e a s u r a b le function f:X->Y d e t e r m in e s a m e a s u r e f(ju) on Y by f(p)B =/uf'1(B) for each B o re l s e t В in Y.

L e t L 0(X,/u;Y) denote the sp a ce of equ ivalence c l a s s e s of m e a s u r a b le f:Y-* Y. A m e t r i c d on Y m a k e s L °(X ,p ;Y ) into a m e t r i c sp a c e w here co n v e rgence i s co n v e rgence in m e a s u r e :

fn -* f in m e a s u r e iff for each p os i t ive e

l im /u{x:d(f (x), f(x)) > e} = 0 n—► °° n

If (Y, d) is a com ple te m e t r i c sp a ce L°(X,/u;Y) is a lso com plete .T hroughout we sh a l l be co n s id e r in g r e a l B anach sp a ce s ; H w ill denote

a s e p a ra b le , in f in i te -d im e n s io n a l H i lb e r t sp a ce , with a fixed in n e r p roduct.

151

152 ELWORTHY

В. CYLINDER SET MEASURES, WEAK DISTRIBUTIONS, RADONIFICATION

F o r a B anach sp a ce E le t

âf = j^ (E ) = { T e L(E, F t ): F t is f in i te -d im e n s io n a l , T onto}

i. e. is the s e t of a l l continuous l in e a r su r je c t io n s of E onto so m e f in ite - d im e n s io n a l B anach space . A cy l in d er s e t m e a s u r e on E can then be co n s id e re d a s a fam ily { p T : T e j / } of p robab i l i ty m e a s u r e s цт on FT such tha t g iven a com m uta t ive d ia g ra m

we have /us = rST(/uT).

E x am p les :(i) G iven a m e a s u r e ц on E define ц? on F? ky = T(p). By abuse

of language a c y l in d e r se t m e a s u r e obta ined th is way is s a id to be a m e a s u r e .

(ii) On H th e r e is a canon ica l fam ily { y T}TC 0< s < o o of cy l inder se t m e a s u r e s . E ac h 7 j is given on a B o re l s e t В с F T by

yr(B) exp (" ^в

w h e re n = d i m F T, | | is the n o rm com ing f ro m the in n e r p ro d u c t induced on F T by T, and dx deno tes L eb e sg u e m e a s u r e c o r re sp o n d in g to th is inne r p roduct.

H ere s is ca l le d the v a r ia n c e p a r a m e t e r .A c y l in d e r s e t m e a s u r e { /uT}T on E a llow s one to d is c u s s the in te g ra t io n

of t a m e functions on E: th e s e a r e those f:E->-R which f a c to r i s e

w h e re Т е ¿s' and fT i s m e a s u r a b le . T he in te g r a l of f o v e r E can then be defined to be

IAEA -SM R -11/22 153

It a l so g ives a f in ite ly additive m e a s u r e /3 on the s u b se ts { T ' 1 (В): Т Е У ,В C Fx B o r e l } i. e. the r in g of c y l in d e r s e t s of E , by /2(T_1 B) = juT(B). The c y l in d e r se t m e a s u r e com es f ro m the m e a s u r e /u on E iff ц is a countably add i t ive ex tens ion of /л.

A nother way of in te g ra t in g ta m e functions on E is by m e an s of a weak d is t r ib u t io n . T h is is an equ iva lence c l a s s of l in e a r functions

а : Е * - Ь ° ( П , р )

of the dual of E into the sp a c e of r e a l - v a lu e d m e a s u r a b l e functions on a p ro b ab i l i ty sp a c e (fi ,p ) . If «в :E * -*L ( ^ ’j P 1) then a is equivalen t to a' iff fo r ea ch finite s e t j i j , . . . , l n} С E* the m aps

a(jfx)X . . .X a(jen) : i 2 - Rn

and

a 'U j J X . . . X a ' ( i n): r2 '^ R n

induce the s a m e m e a s u r e s on R n. U sing such m a p s i t is c l e a r th a t a weak d is t r ib u t io n d e te r m in e s a c y l in d e r s e t m e a s u r e . The c o n v e rse follows f ro m g e n e ra l r e s u l t s on p ro je c t iv e s y s t e m s of m e a s u r e (e. g. in Ref. [13]), so the two notions a r e equivalen t. A continuous l in e a r m ap S : E - > G of B anach sp a c e s sends a cy l in d e r se t m e a s u r e f M t^t on ® in t° a cy l inder s e t m e a s u r e {S (jlí)T} on G by se t t in g S ( j u ) T = p ToS:

The g e n e ra l p ro b le m of rad o n if ic a t io n is to c h a r a c t e r i z e those S for which { S (p )T} is a m e a s u r e on G (usually for a c l a s s of {juT} on E). This has p roved a f ru i t fu l s o u r c e fo r study by functional an a ly s t s [13], [14]; how ever h e r e we sh a l l only d i s c u s s the s i tu a t io n w here { /ux} is one of the canon ica l cy l in d er s e t m e a s u r e s on H.

C. ABSTRACT WIENER SPACES

An a b s t r a c t W iene r sp a ce is a t r ip le (i, H, E) w here i: H-* E is a continuous l in e a r in jec t ion of H onto a d ense su b se t of a B anach sp a ce E such th a t the induced cy l in d er s e t m e a s u r e s { í (ys )t } t on E a r e m e a s u r e s (it su f f ice s to v e r i fy it fo r s = 1). T h ese w e re o r ig in a l ly defined, by G ro s s , in t e r m s of the condition (*) in the th e o re m below. M ost of the b a s ic r e s u l t s which follow a r e a lso due to G ro s s [8, 9].

154 ELWORTHY

A continuous l in e a r in jec t ion i :H -*E with d en se ra n g e d e te r m in e s an a b s t r a c t W iener space iff: (*) G iven e > 0 th e r e is a finite r a n k orthogonal p ro jec t io n P0 on H such th a t fo r any such p ro jec t io n P o r thogona l to Po we have

7 p { x e P ( H ) : H ix К > e> < 1 - e

P r o o f

T heorem (G ro ss , Kallianpur):

We w ill sk e tc h the p roof of the " if" p a r t only: th is i s o r ig in a l ly due to G ro s s but we follow K a l l ia n p u r 's m ethod. A p ro o f of the co n v e rse is g iven in Ref. [10].

L et ¿^"denote the ne t of fin ite d im e n s io n a l o r thogona l p ro je c t io n s on H and l e t a: H*-» L ° ( f2 , p) be a r e p r e s e n ta t iv e of the canon ica l weak d is t r ib u t io n on H co r re sp o n d in g to { y t } . F o r each P in SMve have a m e a s u r a b le m ap

Фр E

defined a s follows: P :H -^P (H ) can be w r i t te n asП

P(x) = ^ (x) e .

i= 1

w h ere е 1л . . . , en is an o r th o n o m a l b a s i s in P(H) and í j e H*, Set

Qp(u) = i a(H; )(co) e ¡^ . The n e t ¿“"converges to I, the iden ti ty on H. i = 1

F o r su ff ic ien tly l a r g e P , Q in ^ w e m ay w r i te P = P 0 + P 1;, Q = Q o + Q i- Condition (*) is then e a s i ly s e e n to be p r e c i s e ly w hat is n eeded to show {Qp}pe grC P ° (Q, p; E) is a C auchy net in m e a s u r e . Thus {Q P }peSr co n v e rg es in m e a s u r e to so m e Q e L° (П , p; E). The m e a s u r e Q(p) on E is then the r e q u i r e d m e a s u r e which induces { í Íy 1) -}.

E x am p les :(i) L e t C 0 (Rn) denote the u s u a l Banach sp a ce of continuous cr:[0, 1] -► R n

with a(0) = 0 and le t L o ’ (Rn) denote i t s in te r s e c t io n with the L 2 ,1 functions. Then Lo ,:L(Rn) i s a H i lb e r t sp a ce with in n e r p roduc t

l

< f , g > = J <f '( t) , g'(t) > dtо

and the in c lu s io n i: L 2, 1 (Rn) -> C 0(Rn) g ives an a b s t r a c t W iener space .T h is i s known as c l a s s i c a l W iene r s p a c e : the induced m e a s u r e 7 1 onC 0 (Rn) is W iener m e a s u r e and is the m e a s u r e c o r re sp o n d in g to B row nian m otion on R n.

The c o r re sp o n d in g r e s u l t holds for the inc lu s ion i : L 2,r (g)->C(f) of sp a c e s of se c t io n s of a R iem a n n ian v e c to r bundle ir: f -*M o v e r a com pac t

smooth manifold M, provided r >

IAEA -SM R -11/22 155

(ii) When E is a H i lb e r t sp a ce and i :H -* E a continuous l in e a r dense in jec t io n then (i, H, E) is an a b s t r a c t W ien e r sp a ce iff i is H i lb e r t -S c h m id t [12].

B a s ic P r o p e r t i e s of an a b s t r a c t W iener sp a ce (i, H, E).

1. E ac h y s is s t r i c t l y pos i t ive : i. e. ea ch open se t in E has p osit ive m e a s u r e .

2. The m ap i is com pac t. F u r th e r m o r e i t s ad jo in t g ives a m apj: E*-* H which is a lso a co m pac t l in e a r in jec t ion with d en se r a n g e and for any T e L(E, E s,s) the m ap jT i :H - » H is t r a c e c l a s s . T h is w as p roved by V. Goodman, an unpub lished p ro o f is in K uo 's th e s i s [11], and it depends on the r e s u l t th a t || | |E is y 1 sq u a re in te g ra b le [7].

Thus we have a m a p t r : L(E , E*)->R, t r T being defined as the t r a c e of jT i . M o re o v e r , if L W(E) deno tes the sp a c e of those m a p s T: E-»E of the fo rm T = I + a , i . e . Tx = x + i j a ( x ) , w here » e L(E, E*) we have a d e te rm in a n t

det: L W( E ) ---- ► R

defined by det(I+o?) = e x p t r lo g ( I+ a ) .L oose ly de tT = d e t T |H c o n s id e r in g t |H : H —H.

3. The m e a s u r e 7 s is q u a s i - in v a r ia n t u n d e r t r a n s l a t i o n by an e le m e n t x of E (i. e. 7 s and i ts im age u n d e r t r a n s l a t i o n by x have the s a m e null se ts ) i f f x e i ( H ) , [22].

4. (see Ref. [33]). The m e a s u r e 7 s is q u a s i - i n v a r ia n t u n d e r the ac tion of an i s o m o rp h is m T e L ( E ) iff T(i(H)) с i(H) and th e induced m ap т |Н :Н -> Н has p o la r decom posi t ion

T |H = U ( l + e )

w h ere U: Н-» H is o r thogona l and a: H-> H is H ilb e r t -S ch m id t .5. A s t r i c t ly pos i t ive m e a s u r e p on a s e p a r a b le B anach sp a ce E com es

f ro m an a b s t r a c t W iener sp a c e (i, H, E) iff p is G a u ss ia n i. e. iff i(p ) has a G a u ss ia n d is t r ib u t io n on R fo r each l e E*.

T h is was p roved by Sato [12] and by K uelbs . The p roof ske tched below w as to ld to m e by Stefan; s e e a l so Ref. [5] fo r a g e n e ra l d isc u ss io n . M ethod of P roo f : Suppose p is a G a u ss ia n m e a s u r e on E. Im m ed ia te ly ,th e r e is a l in e a r m ap E*-* L °(E , p). Since p is G au ss ian it follows ea s i ly th a t th is m aps into L 2 (E, p). U sing the s t r i c t pos i t iv i ty of p we ob ta in a l in e a r m ap

T: E * - L 2 (E, p)

which is in jec t ive .L e t H denote the c lo s u re of T(E*) in L 2 (E, p). This is a H i lb e r t sp a ce

and T: E* -*H. The G a u ss ia n c h a r a c t e r of p show s th a t T is continuous in the w ea k -* topo log ies and hence T is the ad jo in t i* of som e m ap i: H*-» E; th is is the only t r ic k y p a r t in the p ro o f and is not needed if E is a s s u m e d re f lex iv e . It is e a sy to check th a t i induces the m e a s u r e p as r e q u i r e d .

The r e s u l t is e a s i ly ex tended to G a u ss ia n m e a s u r e s which a r e not s t r i c t ly p osit ive and the a b s t r a c t W iener sp a ce s t r u c t u r e can be shown to be e s s e n t i a l l y unique.

156 ELWORTHY

6 . The im age i(H) is a B o re l s e t in E and has 7 s m e a s u r e z e ro for each s . (See exam ple (iii) in sec t io n D below. )

D. CYLINDER SET MEASURES VERSUS ABSTRACT WIENER SPACE

F r o m the defin ition of an a b s t r a c t W iene r sp a ce (AWS) it is c l e a r tha t i t is the canon ica l cy l inder s e t m e a s u r e on H w hich co n t ro ls i ts m e a s u r e th e o re t i c p r o p e r t i e s . T h is is e m p h as ized by the r e s u l t of G ro s s [9] tha t if (i, H, E) is an a b s t r a c t W iener space th e re is ano ther AWS (i1, H, E 1) with a f a c to r iz a t io n

w h e re k: E ' ^ E is com pact. Thus E is ju s t one of m any l a y e r s o f c lo thes which m ake the cy l in d e r s e t m e a s u r e into a m o r e fa m i l ia r - lo o k in g ob jec t for m a th e m a t ic ia n s . In th is s e n se , the b e s t r e s u l t s w ill involve E a s l i t t le as p o ss ib le and hypo theses such as sm o o th n ess with r e s p e c t to E a r e r a t h e r u n n a tu ra l and m ay not be expected to be ob ta ined u s in g m e a s u r e - t h e o re t i c c o n s t ru c t io n s e. g. sm ooth ing o p e r a to r s [19].

The s ta n d a rd m ethod of w ork ing with cy l inder s e t m e a s u r e s is to no tice th a t s ince we can in te g ra te ta m e functions we should t h e re fo r e be able to in te g r a te su i ta b le l im i t s of t a m e functions [ 8].

A s im p lif ied exam ple of th is can be d e s c r ib e d as follows. Choose som e AWS (i, H, E) and an o r thonom al b a s is { e i } " =1 fo r H ly ing in j(E*).The p ro je c t io n s P n of H onto Span { ei, . . . , e n} extend to p ro je c t io n s on E which m ay be co n s id e re d a s m a p s P n : E -* H. Given f: H -> R define f n: E ^ R by fn = f°Pn . P o s s ib ly , {f n} m ay conve rge in m e a s u r e to a m e a s u ra b le f : E ^ R, defined a lm o s t e v e ry w h e re . If so, we can define the in te g ra l of f o v er H to be the in te g ra l of T o v er E. L e t £ denote the l in e a r sp a ce of functions f on H fo r which such an f ex is ts . We then have a l in e a r m ap a: /->• L (E, 7 s), a(f) = f ; (it tu r n s out tha t the choice of s is i r re le v a n t) .In som e se n se f is an ex tens ion of f to a function defined a lm o s t ev e ry w h e re on E: but s in c e i(H) has m e a s u r e z e ro in E the no tion of 'e x te n s io n ' has to be given a m o r e fo rm a l m eaning .

E x am p les :(i) It is ea sy to check th a t H*C-/ so a l in e a r function on H d e te r m in e s

a m e a s u r a b l e function on E. M o re o v e r a | H*: H*-> L° (E, 7 s ) is a r e p r e s e n t a t ive of the weak d is t r ib u t io n on H co r re sp o n d in g to the cy l in d er s e t m e a s u r e

iH E

E '

2 1(ii) F o r c l a s s i c a l W iener sp a ce co n s id e r f: L 0‘ (R)-*R given by

1

0

IAEA -SM R -11/22 157

for som e fixed g e L 2 . T h is could a l so be w r i t te n as a S t ie l t je s in te g ra l

l

f(o-) = J g(t) da (t)о

But s in c e a lm o s t a l l functions in C 0(R) have unbounded v a r ia t io n f s t i l l does not have a c l a s s i c a l m ean ing a lm o s t e v e ry w h e re on C0 (R). H ow ever, f e / and a m e an ing can, and is , a s s ig n e d to it a s a m e a s u r a b le function on C0 (R); see , e .g . Ref. [16].

This is r e a l ly ju s t a sp e c ia l c a se of exam ple (i), but fo r m o r e co m p l i ca ted in te g ra n d s g(t, a(t)) we a r e led to s l igh tly d if fe ren t m e thods of ex tens ion : the s o - c a l le d s to c h a s t ic i n t e g r a l s . T h e r e is a whole ca lcu lu s , developed by Ito [17] and with v a r io u s m od if ica t ions [18], to dea l with such s i tu a t io n s . This highly s o p h is t ic a te d m a th e m a t ic s is m uch u se d by en g in e e rs , fo r exam ple in the co n tro l of m o d e rn c h e m ic a l p lan ts .

(iii) [ 8 ] Define f: H-> R by f(x) = e " lx 2 . Then

f f nM d 7 S(x)= / e |x|Z e 2s dxT Rn

= (2 s + l ) " n/2

Thus fn -* 0 in L 1 (E, 7 s ). Hence, f e S ^ a n d f is iden t ica l ly z e ro although f w as s t r i c t ly pos it ive . It follows in p a r t i c u l a r tha t the m ap a i s not in jec t ive .

T h is exam ple can a lso be u se d to show tha t i(H) has 7 s m e a s u r e z e ro by noting tha t , s in c e fn -» 0 in m e a s u r e , so m e subsequence of { fn}must converge to z e ro a lm o s t e v e ry w h e re [1]. H ow ever, fn -> f po in tw ise on H.

E x am p le (iii) shows tha t one of the m a in p ro b le m s in th is ap p roach is to c h a r a c t e r i z e a c l a s s ^ of functions such t h a t o ' l ^ ' , o r som e m o d if ic a t ion of a \ÿ , b e c o m e s in jec t ive . F o r f u r th e r d is c u s s io n see [15, 8 ].F o r m e a s u r e s on m an ifo lds th e r e a r e m o r e obvious p ro b le m s as w ell and it looks a s if the W iener m anifo ld th e o ry d e s c r ib e d below, which w orks so w ell in the a b s t r a c t , should be c o n s id e red only a s a f i r s t ap p ro x im atio n which hopefully g ives som e g e o m e t r i c a l in tu i tion to what can ac tua lly happen (see s e c t io n H). P o ss ib ly , i ts r e la t io n s h ip to co n c re te s i tu a t io n s is l ike tha t of what A rnold c a l ls the P r o c r u s t e a n bed of the th e o ry of Banach m an ifo ld s to c o n c re te a n a ly s is , which te n d s to o cc u r on the l e s s t r a c t ib le F r e c h e t m an ifo ld s of C ” functions.

E. TRANSFORMATION O F INTEGRAL FORMULA

F o r a fixed a b s t r a c t W iene r sp a ce (i, H, E) and an open se t U of E a m ap f: U-> E will be ca lled a C1 W (I)-m ap if it has the fo rm f(x) = x + a (x ) w here o :U -> E* is C r. (H ere , and subsequen tly , j: E*-> H, to g e th e r with i: H -> E a r e u se d to identify E* and H as su b s e ts of E. ) The following

158 EL WORT H Y

is s l igh t ly w ea k e r than K uo 's o r ig in a l th e o re m but addi tiona l co n s id e ra t io n s have allowed us to d rop h is a s su m p tio n tha t E* is s e p a ra b le .

T h e o re m (H-H. Kuo [11]). L e t Q: U -* V be a C 1 W (I)-d if feom orph ism betw een open su b s e ts of E. Define gs (Q, -): U -» R by

- 2 < Q(x) - x, x ) - I Q(x) - x f(Q, x) = I det DQ(x) | exp -j —

Then

J f(y) dy(y ) - J f°Q(x) g s(Q ,x)dY S(x) v и

for any m e a s u r a b le f: V -* R which m a k e s e i th e r s id e ex is t .

Note : The n o r m in H is denoted by | | to d is t ingu ish it f ro m the n o rm"["j f]” on E. The no ta tion ( , У r e f e r s unam biguously to the p a i r in g of E* with E and to the in n e r p ro d u c t of H.

The th e o re m can be r e p h r a s e d to say th a t Q '1 (-ys | V) % y s | u and the-1 / S \

R adon-N ikodym d e r iv a t iv e — = g s (Q, “).

P . ABSTRACT WIENER MANIFOLDS

We define a Cr A b s t r a c t W ien e r m anifo ld (AWM) o v er (i, H, E) to be a Cr m anifo ld M m ode lled on E with an equ iva lence c l a s s of a t la s e s { (U i.Q jH i, Q¿: Uj -»E such tha t e a c h Q j o Q j '1 is a CrW (I)-map. T h is will a lso be ca lled a Wr~ s t r u c tu re on M. T h is is a m in o r m odif ica tion of K uo 's definition.

Since the t r a n s i t io n m a p s Q¡ ° Qj:1 of an AWM r e s t r i c t to d i f fe o m o rp h ism s on the in te r s e c t io n s of t h e i r dom ains with H and with E* the c h a r t s {(U¡, Q i )}i r e s t r i c t to give m anifo ld s t r u c t u r e s to the su b s e ts MH =u Qi-1 (H),

-1 * 1 M E# = U Q¡ (E*) of M; the m ode l sp a c e s being H and E, r e sp e c t iv e ly ,

iM o re o v e r , (for г й 1), applying the s a m e a rg u m e n ts to the d e r iv a t iv e s ofthe t r a n s i t io n m a p s , we have bundles m ode lled on H and E*, H(M) -» M,E*(M) - M with E*(M) С H(M) С T(M). L e t E ;(M ) — ► HV(M ) . ---- ► T Mjx x ix x

denote the in c lu s io n s of the f ib re s o v e r a point x of M. Note th a t H(M) r e s t r i c t e d to M H is ju s t TMH and s im i l a r ly for E*(M) r e s t r i c t e d to M E*. P r o p e r ty 2 of s e c t io n С show s th a t a Wr s t r u c t u r e , r è 1, d e t e r m in e s a F re d h o lm s t r u c t u r e on M [26, 25], and on M E*, and even a "n u c le a r s t r u c t u r e " on M H. In p a r t i c u l a r i t d e te r m in e s an e lem e n t of KO(M) [26].

F o r a f ixed v a r ia n c e p a r a m e t e r s > 0 and an AWM as above define

Sij = g*j : U i n Uj - GL(R) = R - { 0 }

by

g¡j (x) = gs (Qj Q j S Q í (x))

IAEA-SM R-11/22 159

P r o p e r t i e s of R adon-N ikodym d e r iv a t iv e s im m e d ia te ly im ply

gy (x )g jk(x) = g lk (x) x G Uj П Uj (1 U,

Hence the fam ily { g ^ j j j fo rm the t r a n s i t io n functions for a l ine bundle Ж(М) o v e r M [27].

T h is w ill be ca l le d the bundle of W ien e r d e n s i t ie s o v e r M and i ts s e c t io n s w il l be ca l le d W iener d e n s i t ie s on M (v a r ia n c e p a r a m e te r s).Such a dens ity Ç is d e te rm in e d by a fam ily {?¡}; of functions f¡ : U¿ -*■ R such tha t

gy (x) ?j (x) = fj (x) fo r x £ Uj П Uj

Since each gy is pos i t ive we can say tha t 5 is a pos it ive dens ity if each > 0 . F in a l ly such a pos i t ive d ens i ty Ç d e te r m in e s a s t r i c t ly posit ive

B o re l m e a s u r e ц (Ç) on M by a s s ig n in g

/i(f)(B)= J d7 s(x)

Q¡(B)

to a B o re l s e t B c U ¡ . A cco rd ing to K uo 's th e o re m , th is is independent of the choice of c h a r t con ta in ing В and i t is e a sy to s e e tha t i t does , in fact, extend to a countably additive m e a s u r e as r e q u i r e d .

In s u m m a ry , a Wr s t r u c t u r e , r è 1, on a B anach m anifo ld M d e te r m in e s a unique m e a s u r e c la s s on M fo r ea ch v a r ia n c e p a r a m e te r and the pos i t ive d e n s i t ie s d e te rm in e the m e a s u r e s . In fact, th is is au tom at ic , given a t r a n s f o r m a t io n of the in te g r a l fo rm u la , and does not depend, at a l l , on the ac tua l fo rm of the R adon-N ikodym d e r iv a t iv e s involved (although it is convenien t to have th e m continuous).

In f in ite d im e n s io n s , th e re is a g e o m e t r i c a l way of ob ta in ing d e n s i t ie s , n am ely by m e a n s of a R iem a n n ian m e t r i c [2] and th is is p a r t i c u la r ly con v en ien t b e c a u s e i t a lso induces d e n s i t ie s on subm anifo lds ; so, fo r exam ple , subm anifo lds of Rn have canon ica lly defined m e a s u r e s on them . In our c a se s l igh tly m o r e is needed b ec au se of the la ck of t r a n s l a t i o n in v a r ian ce of our b a s ic m e a s u r e s 7 s. A Cx W ie n e r -R ie m a n n ia n m e t r i c G on an a b s t r a c t W iener m anifo ld M is defined to be a R iem ann m e t r i c ^ / x on the bundle H(M) o v e r M such tha t in a c h a r t (Ui ( Qi) the lo c a l e x p r e s s io n fo r u , v ) x can be w r i t te n as <^Gxu, GXV / w h ere Gx e L w(E) for each x in Qj(Ui) and x Gx : Qi(Ui) - L W(E) is Cr .

Note th a t fo r each x in M the choice of such a G m a k e s the inc lu s ion i x : H X(M) -► TXM into an a b s t r a c t W ien e r sp a ce ( in it ia l ly HX(M) was ju s t H i lb e r ta b le and th e defin ition of an AWS involves a p a r t i c u l a r in n e r p roduct) ; m o r e o v e r , the m ap j x: E* (M) -> Hx (M) is th a t obta ined f ro m the adjoin t of i x. In p a r t i c u la r , E* (M) b e c o m e s iden tif ied with th e co tangent bundle T*M of M s ince the R iem a n n ian m e t r i c ex tends to a g lobal p a i r in g of E*(M) with TM.

F o r a d ens i ty we sh a l l a l so need a pos i t ion f ie ld , Z o n M . T h is is a v e c to r f ield Z: M -> TM such th a t in a c h a r t (Q¿, U¡ ) th e p r in c ip a l p a r t of Z,Z 1: Qi(Ut ) -» E is a W (I)-m ap. A p a i r (G, Z) c o n s is t in g of a WE m e t r i c and

160 ELWORTHY

a pos i t ion field w ill be ca lled W iene r da ta on M. F o r such data and for each c h a r t (Q¡, U¡) of M define

by

p. (x) = I det Glv I exp -j - —

P¡ = p ¡ ( G , Z ) : Q . ( U j ) - R

2 < G l Zi ( x ) - x , x > + I x - G^Z'fx) |2

T his can e a s i ly be checked to be the lo c a l e x p r e s s io n of a p o s i t iv e dens ity p s(G, Z) on M, v a r ia n c e p a r a m e t e r s. Thus, W iene r data on M d e t e r m in e s a m e a s u r e /js = ns (G, Z) on M fo r a l l 0 < s < oo.

The d e n s i t ie s ps(G, Z) a l so have a m o r e g e o m e tr ic a l desc r ip t io n .W ien e r da ta (G, Z) induce m e a s u r e s 7 X on each tangen t sp a c e T XM, yx being defined a s the t r a n s l a t e of the a b s t r a c t W iener m e a s u r e on Tx M, d e te rm in e d by (ix , H X(M), TXM), by the e le m e n t Z(x) of T XM. O v er a c h a r t QjfUi), fo r x in QjfUj) the m ap

v « x + v : T x M - » E

sends 7 x to a m e a s u r e on E which is equivalen t to 7 s . K uo 's th e o re m gives an e x p r e s s io n for the R adon-N ikodym d e r iv a t iv e , and i t s va lue at x is p r e c i s e ly p- (G, Z)(x). Thus, in som e se n se the m e a s u r e s 7 X a r e ' tangen t ' to the m e a s u r e s / / . A no ther p ro p e r ty of the m e a s u r e s p s (G, Z) is th a t as s -> 0 they conve rge , at l e a s t in som e loose se n se , to som eth ing c o n c e n t ra te d on the z e r o s of Z; p oss ib ly th is could have in te re s t in g app l ica t ions ( see expam ple (iii) below).

E x am p les :

(i) An open s u b se t of E has a t r i v i a l W ” s t r u c t u r e and can be given the s t a n d a r d W iener d a ta co n s is t in g of the given in n e r p ro d u c t on H and th e v e c to r f ie ld Z with Z(x) = x. Then p s is ju s t 7 s i ts e lf .

(ii) If M is Cr B anach m anifo ld , г й 1, and f: M -► E is a Cr F re d h o lm m ap of index z e r o then f induces a uniquely defined W1 s t r u c t u r e of M.T his follows f ro m the co r re sp o n d in g r e s u l t about s t ro n g l a y e r s t r u c t u r e s in Ref. [24]. M o re o v e r , when f i s p r o p e r i t is r a t h e r ea sy to obta in an in te g ra l fo rm u la for th e d e g re e of f u s ing th is s t r u c t u r e [6 ]. F r o m Ref. [25] (or [26], if r s 3) it follows th a t if M ad m its C r p a r t i t io n s of un ity and is p a r a l l e l i s a b le then it ad m its at l e a s t one W r s t r u c t u r e for ea ch e lem e n t of KO(M). The r e la t io n s h ip s between the m e a s u r e c l a s s e s co r re sp o n d in g to th e s e s t r u c t u r e s is not c l e a r .

(iii) If E 0 i s a f in i te -c o d im e n s io n a l c lo sed su b sp a c e of E th e r e is a canon ica l sp l i t t ing E = Eq X E 0 w h e re Eq is the ann ih i la to r of E 0 in E*.If H 0 = E 0 П H th e n Eq is a lso the o r thogonal com plem en t of H 0 in H. The inc lu s ion i0 : H 0 -* E 0 i s e a s i ly se e n to be an a b s t r a c t W iene r space .

Suppose tha t M is a Cr subm anifo ld of E of cod im ens ion n. Choose E 0 of cod im ens ion n and l e t f: M -* Eo be the r e s t r i c t i o n to M of the p r o je c t io n of E onto E 0 . Then f is F re d h o lm of index z e ro and we m ay apply(ii) to get a canon ica lly defined W r s t r u c t u r e on M, m ode l led on ( i 0, H0 , E Q). This is e s s e n t i a l ly independent of the choice of E 0. We a lso have W iener d a ta (GM, Z M) induced on M: GM ju s t being the n a tu ra l ly induced m e t r i c

IA EA -SM R-11/22 1 6 1

f ro m th e in c lu s io n of M into E and Z M(x) being defined as the p ro je c t io n of x onto TXM. T hus, any f in ite c o -d im e n s io n a l subm anifo ld of an a b s t r a c t W iener sp a c e has a well defined fam ily { p s } of m e a s u r e s on it.

As a c o n c re te exam ple take c l a s s i c a l W iene r sp a ce L 0’ (Rn) -+ C 0(Rn) and le t M C C ( (Rn) c o n s is t of th o se pa ths which end on a sm ooth su b m an ifo ld Yn"p of Rn. In th is c a s e , the z e r o s of Z M a r e ju s t the c r i t i c a l po in ts of the e n e rg y in te g ra l on M П Lq’1 (Rn). An a l te rn a t iv e p r o c e d u re fo r a s s ig n in g a m e a s u r e to M is g iven in Ref. [6]. This t u r n s out to be the m e a s u r e with den s i ty (27ts)"P//2 exp[ - \ \ x - Z M(x) *] p 1(GM, Z M ); se e a l s o Ref. [21] and the fo rm u la in the d iv e rg e n ce th e o re m below.

G. DIVERGENCE THEOREMS AND THE LAPLACIAN

A v e c to r f ie ld X on an a b s t r a c t W iener m anifo ld M w ill be ca lled a d m is s ib le if i t f a c to rs th rough a s e c t io n of E*(M)-> M. L e t F: D С M X R -* M be the flow of such a v e c to r f ield X. It is ea sy to se e tha t each m ap F t = F ( - , t ) is a W (I)-m ap, i. e. is W(I) in the c h a r t s of M. If Ç is a posit ive den s i ty on M with co r re sp o n d in g m e a s u r e p it follows by K uo 's t h e o r e m tha t F t (p | D П MX t) is ab so lu te ly continuous with r e s p e c t to p. M o reo v e r , we can define the d iv e rg e n ce of X with r e s p e c t to Ç by

We can now give the s im p le s t d iv e rg e n c e th e o re m :

P ro p o s i t io n : L e t X be a com ple te ly in te g ra b le a d m is s ib le v e c to r f ie ld onan a b s t r a c t W iener m anifo ld M, of c la s s C 1. T hen fo r any pos it ive W ien e r dens ity Ç on M with a s s o c ia te d m e a s u r e p such th a t p(M)<oo we have

/ Div X d/i = 0

p ro v id ed D iv jX G L 1(M,p).Since m a th e m a t ic ia n s tend to l e a r n the f in i te -d im e n s io n a l d ive rgence

th e o re m as a c o r o l l a r y of the m uch m o r e com plica ted S toke 's th e o re m it s e e m s w orthw hile pointing out the e s s e n t i a l t r iv ia l i ty of the proof:

Schem e of proof: F o rm a l ly :

(p))divE X dp_d_ dÇF dt dp

d

t = о M

d (F . t (p)) d p dp

t=0

■ ж I

= f , J ■ I “ |M|Ft(M)

= 0

the only p ro b le m being to ju s t i fy the d i f fe ren t ia t io n u n d e r the in te g r a l sign.

1 6 2 ELWORTHY

F o r the ca se of a dens ity a r i s in g f ro m W ien e r d a ta (G, Z) we can find a p a r t i c u la r ly n ic e exp lic i t e x p r e s s io n fo r the d iv e rg e n ce . The R iem ann m e t r i c induced by G on H(M) and thence on T M H has a L e v i -C iv i ta connection which can be shown to extend to a connection on the whole of TM with connection m ap l> : TTM ->■ TM h a v in g the lo c a l fo rm , [28], over (Qi.U i) , f> (x, Ç, y, rj) = (x, r] + l>i (x)(y, ?)) w h e re F>¡ : Q j(Uj ) - L 2 ( E X E , E*).L et V denote c o v a r ian t d if fe ren t ia t io n with r e s p e c t to th is connection.Then if X is C 1 a com puta t ion show s tha t

Div X(x) = t r V X(x) - — < VZ(X), Z>ь g x

w h ere Divs r e f e r s to the d iv e rg e n ce with r e s p e c t to ps (G, Z). F o r s ta n d a rd W iene r da ta on E th is r e d u c e s to

Div X(x) = t r DX(x) - - < X ( x ) , x >“ s

To have m o r e u se fu l hypo theses for the d iv e rg e n ce and to d ea l with m an ifo lds with boundary we sh a l l need the notion of c o m p le ten e ss of W -R m e t r i c s . We have shown how M contains a H i lb e r t m anifo ld M H, but in fac t each point x of E is contained in the c o se t x + H of H and th e se co se ts m ay equa lly w ell be pulled back u s in g the c h a r t s of M to obta in H ilb e r t m an ifo lds M h(x) with in c lu s io n s M h(x) M such tha t each point of M l ie s in one such 'c o s e t ' m anifold . A W -R m e t r i c G on M m a k e s each of th e se M H(X) into a R iem ann ian m anifo ld , and we say G is com ple te if th e se a r e a l l com ple te in the u s u a l se n se . I do not know w hethe r th is is equ ivalen t to the geodes ic c o m p le te n e s s of the connection induced by G on M.

A s i m i l a r r e s u l t to the following, but fo r dom ains in E , has been ob ta ined by Skorohod [20] and Goodman [19]. G oodm an 's r e s u l t has in te re s t in g ly w ea k e r hy p o th e ses which a r e p a r t i c u la r ly r e le v a n t to the r e m a r k s at the end of s e c t io n H below.

D iv e rg e n ce th e o re m

L e t M be an a b s t r a c t W iener m anifo ld with boundary 3M (possib ly 3M= 0) with C 3 W iener da ta (G, Z). A ssu m e G is com plete . L e t X be an a d m is s ib le C 1 v e c to r f ield on M such tha t the m ap x -* | x ( x ) | x is in L 1(M ,ais (G, Z)). Then

j DivsX d,us(G,Z) = -j===J< n ( x ) , X ( x ) > e x p - 1 ^ - <n(x), Z(x)>2| d/js (GeM , Z 0M )M M

prov ided both in te g r a l s ex is t , and w h ere n(x) is the uniquely defined in te rn a l n o r m a l to 3M a t x with | n ( x ) |x = 1.

The b a s is of the p ro o f is to w r i te X = X j + X 2 w here X j is tangen t to ЭМ and X2 is n o r m a l to ЭМ and has su p p o r t in a c o l la r neighbournood of ЭМ. We can modify the s im p le d iv e rg e n c e th e o re m to show tha t the in te g ra l of the d iv e rg e n c e of Xj v a n ish e s , and can u s e F u b in i 's th e o re m to eva lua te the in te g r a l of the d iv e rg e n c e of X 2 . The r e s u l t ex tends v a r io u s c l a s s i c a l th e o re m s fo r ' in te g ra t io n by p a r t s ' in c l a s s i c a l W iener sp a ce [30].

IA EA -SM R-11/22 163

F o r a С 1 m a p f:M -»R on an AWM with a W -R m e t r i c we can define g ra d f to be the a d m is s ib le v e c to r f ie ld d e te rm in e d by the se c t io n df of E*(M) = T(M)*. Under the hy p o th e ses of the d iv e rg e n c e th e o re m but with ЭМ = f> we have the é o r o l l a r y th a t Divs a n d - g r a d a r e fo rm a l ly adjoint o p e r a to r s . If f i s C2 and we define the L ap la c ia n A s of f by

Asf = Div g rad f

it follows tha t Д 5 is fo rm a l ly se l f - a d jo in t and nega t ive defin ite . In p a r t i c u la r if A s f = 0 and f is С 2 and a lso f £ L 2 (M, /u2 ) with | g rad f |x e L 2 (M, ц* ) i t a lso follows tha t f is constan t . Note th a t As f h a s the e x p re s s io n A s f = t r V 2 f - ( l / s ) ‘\ V Z ( g r a d f), Z)> which for M = E with s ta n d a rd W iener da ta r e d u c e s to

As f(x) = t r D2 f(x) - ^ Df(x)x

T h is L ap la c ia n on E has been s tud ied , in a s l igh t ly d if fe ren t context, by U m e m u ra [23]. In p a r t i c u la r , i ts e igenva lues and e ig e n sp a c e s can be found: the e ig e n v e c to r s a r e F o u r i e r - H e r m i t e po lynom ials . A pparen t ly , it can a lso be a s s o c ia te d with a h a r m o n i c - o s c i l l a t o r H am ilton ian in quantum field theo ry .

The L ap la c ia n Д „ defined by A J = t r D2f has been s tu d ied ex tens ive ly by G ro s s [22]; s e e a lso Ref. [21]. F o r m o r e g e n e ra l r e s u l t s on an a ly s is on in f in i te -d im e n s io n a l s p a c e s , s e e Ref. [35] and i t s b ib liography . A s to k e s 1 th e o re m u s in g "f in ite co -d im e n s io n a l d i f fe ren t ia l f o rm s " is in p r e p a ra t io n by R. R a m e r (A m ste rdam ).

H. BROWNIAN MOTION ON A COMPACT MANIFOLD;CONCLUDING REMARKS

F o r a com pac t R iem a n n ian m anifo ld X le t CXo (X) denote the Banach m an ifo ld of continuous paths a : [0, 1] -*X with a (0) = x 0. L e t L XJ (X) denote the c o r re sp o n d in g H ilb e r t m anifo ld of L 2,1 pa th s [28]. T h e re is a n a tu ra l B row nian m otion m e a s u r e w on C Xo ob ta ined u s in g th e fundam enta l so lu t ion of the h ea t equa tion on X, [6]; s e e a l so Ref. [32] fo r the r e la t io n sh ip s be tw een m e a s u r e s , d iffus ion p r o c e s s e s , and p a ra b o l ic equations . It i s 'n o t known w hethe r o r not the m anifo ld Cx ad m its a n a tu ra l Wr - s t r u c t u r e (or even if it ad m its one a t all). H ow ever, th e r e is a d if feo m o rp h ism , the C a r ta n deve lopm ent, &): L 2,1(TX() X) -> L ^ t X ) , [6 ,29] . This does not extend to a m ap of C 0 (TXo M) nor even to one with dom ain any of th e sp a c e s of H older continuous functions which have, full c l a s s i c a l W iene r m e a s u r e in C 0(TXoX). H ow ever, we can p ro c e e d as in s e c t io n D and choose p ro jec t io n s Pn which m a p C 0(TXoX) onto the f in ite d im e n s io n a l su b sp a ce s spanned by the v e c to r s of an o r th o n o m a l b a s e of L2,1 (TX[l X) co n s is t in g of p ie ce w ise l in e a r functions. T h is way we obta in m a p s 2&° Pn : C0 (TXo X) - L x' (X) which a r e e a s i ly se e n to f a c to r i s e as

HL ц’ \р (Х )) c CUo(P(X))

s I 7Г I 7Г

L 2 ll (X) С С (Р(Х))

164 ELWORTHY

fo r a su i tab le choice of Q n (which can be exp l ic i t ly w r i t te n in t e r m s of exponen tia l m ap s) , w h e re H L2;1 (P(X)) deno tes the space of h o r iz o n ta l L 2,1 pa th s in the p r in c ip a l bundle P(X) of X s ta r t in g f ro m a fixed f r a m e u0 in TXo (X), and w h ere ît deno tes the n a tu ra l p ro je c t io n s . It follows f ro m a (very n o n - t r iv ia l ) r e s u l t of Gangoll i [31] th a t the m a p s Q n : C 0(TXo X) ->• Cu (P(X)) co nve rge a lm o s t e v e ry w h e re with r e s p e c t to c l a s s i c a l W ien e r m e a s u r e y 1 to som e Q: C0 (TXo X) -» CUo (P(X)) and th a t the m e a s u r e îrQfy1) is p r e c i s e ly the B row nian m otion m e a s u r e w.

This im p o r ta n t exam ple g ives u s a m e a s u r e w on a Banach m anifo ld CXo (X) which i s tan ta l iz in g ly c lo se to f it t ing into o u r a b s t r a c t th e o ry , but ap p a ren t ly does not. P o s s ib ly w o rs e , we a lso have a m e a s u r e Q fy1) on the c lo s u re in CUo (P(X)) of HL u2'-'(P(X)) and th is is p e rh a p s not even a m anifo ld , in g en e ra l ; but, n e v e r th e le s s , the m e a s u r e is induced in so m e se n se by the dense H i lb e r t m anifo ld . It s e e m s , t h e r e f o r e , th a t a u se fu l a im would be to t r y to ex tend the a b s t r a c t th e o ry and t r y to find exac t ly what s t r u c t u r e is needed on an in f in i te -d im e n s io n a l sp a ce to ob ta in a g e o m e tr ic a l ly defined m e a s u r e with so m e G a u ss ia n c h a r a c t e r , and, in p a r t i c u l a r , to obta in a b e t t e r u n d e rs ta n d in g of the g e o m e try of th e se B row nian p r o c e s s e s .

R E F E R E N C E S

B a s ic s :

[1] KINGMAN, J. F.C., TAYLOR, S.J., Introduction to Measure and Probability, Cambridge(1966).[2] LOOMIS, L.H., STERNBERG, S., A dvanced Calculus, Addison-Wesley (1968).[3] PARTHASARATHY, K.R., Probability Measures on Metric Spaces, Academic Press (1967).

G enera l :

[4] DUDLEY, R.M., Sample functions of the Gaussian process, M.I.T. (1972).[5] DUDLEY, R.M., FELDMAN, J., LeCAM, L., On seminorms and probabilities, and abstract Wiener

spaces, Ann. Math. 93 (1971) 390.[6] EELLS, J., ELWORTHY, K.D., Wiener integration on certain manifolds, in "Some Problems in

Non-Linear Analysis", Centr. Int. Mat. Est. 4 (1970) 67.EELLS, J., Integration on Banach manifolds, Proc. 13th Biennial Seminar Can. Math. Congress,Halifax (1971).

[7] FERNIQUE, M.X., Intégrabilité des vecteurs gaussiens, C.R. Acad. Sci. Paris, Ser. A, 270 (June 1970)' 1698.

[8] GROSS, L., Measurable functions on Hilbert space, T.A.M.S. 105 (1962) 372.[9] GROSS, L., Abstract Wiener Spaces, Proc. 5th Berkeley Symp. Math. Stat. and Probability 1965/6, 31.

[10] KALLIANPUR, G . , Abstract Wiener processes and their reproducing kernel Hilbert spaces, Z. Wahrschein- lichkeitstheorie 17 (1971) 113.

[11] KUO, H.-H,, Thesis, Cornell 1970, published as: Integration Theory on infinite-dimensional manifolds, T.A.M.S. 159 (1971) 57.

[12] SATO, H . , Gaussian measure on a Banach space and abstract Wiener space, Nagoya Math. J. 36 (1969) 65. (Note the correction in Ref.[5].)

R adonif ica tion , functional ana ly tic a s p e c ts :

[13] S C H W A R T Z , L., Applications Radonifiantes, Séminaire d* Analyse de l'Ecole Polytechnique, Paris (1969/70).S C H W A R T Z , L., Applications Radonifiantes, to appear in J. Math. Soc. Japan, in honour of Yosida (1971).

[14] See also articles in Studia Math. 38.

IAEA -SM R -11/22 165

Weak d is t r ib u t io n s :

FRIEDRICHS, K.O., SHAPIRO, H.N., Integration over Hilbert space and outer extensions, Proc. Nat. Acad. Sci. USA 43_ (1957) 336.

[15] SEGAL, I.E., Algebraic integration Theory, Bull. A m . Math. Soc. 71 (1965).See also Ref.[8] and the works of Segal.

S to ch a s t ic in teg ra t io n :

[16] NELSON, E., Dynamical Systems and Brownian Motion, Mathematical Notes, Princeton University Press (1967).

[17] McKEAN, H.P., Stochastic Integrals, Monographs on Probability and Math. Stats. No.5, Academic Press (1969).

[18] McSHANE, E.J., Stochastic Differential Equations and models of random processes, to appear in Proc. 6th Berkeley Symp. Mathematical Statistics and Probability.

D iv e rg e n ce th e o re m s :

[19] G O O D M A N , V., A divergence theorem for Hilbert spaces, T.A.M.S. 164 (1972) 411.[20] SKOROHOD, A.V., Surface integral and Green* s formula in Hilbert space, Teor. Verojatnost. Mat.

Statist. 2 (1970) 172. (In Russian).STENGLE, G., A divergence theorem for Gaussian stochastic process expectations, J. Math. Anal. Appl. 21 (1968) 537.

P o te n t i a l theory :

[21] G O O D M A N , V., Harmonic functions on Hilbert space, J. Functional Analysis 10 (1972) 451.[22] GROSS, L., Potential theory on Hilbert space, J. Functional Analysis 1 (1967) 123.[23] U M E M U R A , Y., On the infinite dimensional Laplacianoperator, J. Math. Kyoto Univ. 4 (1965) 477.

D if fe re n t ia l topology:

[24] EELLS, J., ELWORTHY, K.D., Open embeddings of certain Banach manifolds. Ann. Math. 91 (1970) 465.[25] ELWORTHY, K.D., Embeddings, isotopy and stability of Banach manifolds, Comp. Math. 24 (1972) 175.[26] ELWORTHY, K.D., TROMBA, A.J., Fredholm maps and differential structures on Banach manifolds.

Global Analysis (Proc. Sympos. Pure Math. 15, Berkeley, Calif., 1968), 45-94 A.M.S. (1970).Nicole M O U LIS: Structures de Fredholm sur les variétés Hilbertiennes, Lecture Notes in Math. 259 Springer-Verlag (1972).

[27] STEENROD, N., The Topology of Fibre Bundles, Princeton University Press (1951).

D if fe re n t ia l g eom etry :

[28] ELIASSON, H.I., Geometry of manifolds of maps, J. Diff. Geom. 1 (1967) 169.[29] KOBAYASHI, S., Theory of connections, Ann. di Mat. 43 (1957) 119.

KOBAYASHI, S., NOMIZU, K., Foundations of Differential Geometry, 1, Interscience (1963).

C la s s i c a l W iener m e a s u r e :

[30] KOVAL'CHIK, I.M., The Wiener integral, Rus. Math. Surveys 18 (1963) 97. See also Refs [13,16,17, 1]. ~

166 EL WORTHY

Brow nian m otion on f in i te -d im e n s io n a l m anifo lds:

[31] GANGOLLI, R ., On the construction of certain diffusions on a differentiable manifold, Z . Wahrschein- lichkeitstheorie 2 (1964) 406.

[32] NELSON, E ., An existence theorem for second-order parabolic equations, T .A .M .S . 88 (1958) 414. See also Ref.[17].

Additional:

DALETSKII, Y u .L ., SHNAIDERMAN, Y a .I . , Diffusion and quasi-invariant measures on in fin itedimensional Lie groups, Functional Anal. Appl. 3 2 (1969) 88.

[33] GUICHARDET, A . , Symm etric Hilbert Spaces and Related Topics, Lecture Notes in Math. 261 Springer-Verlag (1972),

[34] ITO, K . , The Brownian motion and tensor fields on Riemannian manifold, Proc. Int. Congr. M ath .,Stockholm (1963) 536.NEVEU, J . , Processus aléatoires gaussiens, Sém de Math. Sup. M ontreal (1968) .VENTSEL, A .D ., FREIDLIN, M .I . , On small random perturbations of dynam ical systems. Russ. Math.Surveys 25 1 (1970) 1.

[35] VISHIK, M .I . , The param etrix of e llip tic operators with infinitely many independent variables, Russ. M ath. Surveys 26 (1971) 91.

IAEA-SM R-11 /23

SHEAF COHOMOLOGY, STRUCTURES ON MANIFOLDS AND VANISHING THEORY

M.J. FIELD Mathematics Institute.University of Warwick,Coventry, Warks,United Kingdom

Abstract

SHEAF COHOMOLOGY, STRUCTURES ON MANIFOLDS AND VANISHING THEORY.The paper deals with finding significant and computable invariants of structures on topological spaces.

After a brief review of sheaf cohomology theory some characteristic situations in com plex manifold theory are exam ined. Stress is laid upon finiteness and vanishing theorems. The m ateria l is presented in approximately historical order.

In th is p a p e r , we w ish to c o n s id e r the following type of p rob lem :Suppose we have a topo log ica l sp a c e M, to g e th e r with so m e s t r u c t u r e Sôn'M (for ex am ple , S ^m igh t be a d if fe ren t ia b le o r com plex s t r u c t u r e o r the s t r u c tu r e of an ana ly tic o r a lg e b ra ic se t) . How do we study „9"and find s ign if ican t (and com putab le) in v a r ian ts of S'”?

T h is type of p ro b le m has an teced en ts in the study of topology. H ere the p ro b le m is of the fo rm : Given a topo log ica l sp a ce M (no s t r u c tu re ) find topo log ica l in v a r ia n ts of M which can be u se d to study M and d is t in g u ish it f ro m o th e r topo log ica l sp a c e s . Again com pu tab i l i ty (or m anageab il i ty ) is an im p o r ta n t r e q u i r e m e n t . Of g r e a t im p o r ta n c e in the study of topologica l s p a c e s and m an ifo ld s has been the deve lopm ent of a lg e b ra ic topology. H ere p r o b le m s about topo log ica l sp a c e s a r e re d u c e d to e s s e n t i a l ly a lg e b ra ic p ro b le m s in the s tudy, fo r ex am ple , of the cohom ology r in g of a space .Thus the h ighly s y s t e m a t i s e d and pow erfu l m e thods of hom olog ica l a lg e b ra can be applied to topo log ica l p ro b le m s .

H en r i C a r t a n in the 1950 's , following upon e a r l i e r w ork of h is and o th e rs on a lg e b ra ic topology and h o m olog ica l a lg e b ra m a d e the fundam en ta l o b s e rv a t ion tha t the m e c h a n is m s and techn iques of a lg e b ra ic topology could be applied su c c e s s fu l ly to the th e o ry of s h e a v e s . T h rough the w ork of C a r ta n and l a t e r S e r r e , it r ap id ly b e c a m e c l e a r tha t the a lg e b ra ic topology of s h e a v e s , n am ely sh e a f cohom ology, was a too l of g r e a t pow er fo r ana lys ing s t r u c t u r e s on topo log ica l s p a c e s as w e ll a s p rov id ing a n a tu r a l g e n e ra l iz a t io n of the cohom ology th e o ry of topo log ica l sp a c e s .

In th is p a p e r , a p a r t f ro m b r ie f ly r ev ie w in g the th e o ry of sh e a f cohomology, we sh a l l exam ine so m e c h a r a c t e r i s t i c s i tu a t io n s in com plex m anifo ld th e o ry to show the app l icab i l i ty of sh e a f cohom olog ica l techn iques to function theo r e t i c , a lg e b ra ic and d i f fe re n t ia l g e o m e tr ic p ro b le m s . In addition, we sh a l l e m p h a s iz e the g e n e r a l im p o r ta n c e of f in i te n e s s and van ish ing th e o re m s in th is context. In a p a p e r of th is length , m a n y top ics in th is a r e a w ill of n e c e s s i ty have to be om itted ; in p a r t i c u l a r , no d is c u s s io n w ill be m ade of p ro b le m s in the d e fo rm a t io n th e o ry of com plex s t r u c t u r e s o r of the g e n e ra l

167

168 FIELD

p ro b le m of cohom olog ica l r e p r e s e n ta t io n of o b s t ru c t io n to ex tens ion of com plex s t r u c t u r e s on com plex m an ifo ld s . To u n d e rs ta n d the content of the p a p e r , a knowledge of Ref. [8] ( these P ro c e e d in g s ) is r e q u i r e d . I have a lso end e av o u red to a r r a n g e the m a t e r i a l in an ap p ro x im a te ly h i s t o r i c a l fashion: the m a t e r i a l in the l a s t s e c t io n s w ill r e q u i r e som ew hat g r e a t e r m a th e m a t ic a l backg round than tha t in the f i r s t few se c t io n s . T h e r e w ill a lso be a tendency to p r o g r e s s f ro m an in fo rm a l , concep tua l fo rm u la t io n of the m a t e r i a l to a m o r e f o rm a l p ro b le m a t ic p re se n ta t io n . In so m e w ays, th is will c o r r e s p o n d to a p r o g r e s s io n f ro m a s i tu a t io n w h e re quite a lo t is known (Stein m an ifo lds) to a s i tua t ion w here v e r y l i t t le is known (an a r b i t r a r y com plex manifold).

1. REVIEW OF SHEAF COHOMOLOGY

The m o s t n a tu ra l and g e n e r a l way to define the cohom ology of a sh e a f of c o m m u ta t iv e r in g s 5 ^ o v e r an a r b i t r a r y topo log ica l sp a c e M is by the u se of a flabby (or fine) re so lu t io n of the sheaf . H ere , h ow ever , we give a defin ition b a s e d on the Cech c o n s t ru c t io n which is val id only fo r p a ra c o m p a c t sp a ce s (or, a l te rn a t iv e ly , c o h e re n t sheaves):

Suppose, t h e re fo r e , tha t W = {Uj}is I is an open co v e r in g of the topo log ica l sp a c e M.

Let s = ( i q................. i p) e I p+1. We s e t Us = U ^ n ............ Ûip. We define ap cochain of with va lues in to be a function f, which a s s ig n s to ev e ry s £ I p tl , a s e c t io n fs of &\ Us .

In a n a tu r a l way, the s e t of a l l p cochains fo rm a co m m u ta t iv e r in g which we w ill denote by C p( /,gr).

We define a coboundary o p e r a to r <5:CP(<|/, gr)------► Ср+1( ^ , &) forp = 0 , 1 , . . by the u su a l fo rm u la :

p+l

(6f). . = У ( - i ) V / f . . , \ v v i ^ V o "4 "lp+1)

w h e re , f is a p co cha in ,^deno tes o m is s io n and

r. : r A j . „ . , g r \ -----------------------►Г/'U. . ,gr\1 V V - M - ' V i ) )

is ju s t the r e s t r i c t i o n h o m o m o rp h ism . ("Г" always deno tes se c t io n s . )It is then an e a sy e x e r c i s e to check tha t 62 = 0.With th e se defin i t ions out of the way, we m a y define the p - th cohom ology

g roup of the co m plex ICP by

Hpf c j r ) = K e r 6 : C p( ^ , g r ) ----- ^ Cp + 1 fagr)

Im ------.. С

The fac t tha t 6Z = 0 im p l ie s tha t the den o m in a to r of the above e x p r e s s io n is indeed a subg roup of the n u m e ra to r .

IA EA -SM R-11/23 169

If T is a r e f in e m e n t of the c o v e r <%/, then it is not difficult to show that we have a n a t u r a l h o m o m o rp h ism induced f ro m H q (‘/ ' S'") to H4(<g/, Sr ).H ^M , is then defined as the d i r e c t l im i t of the S'") over in c re a s in g lyfine r e f in e m e n ts .

Hq(M ,á /) is ca lled the q - th sh e a f cohomology g roup of M with coe f fic ien ts in W. We w ill s o m e t im e s w r i te it in ab b re v ia te d fo rm H4( ¡P~).

We have the following b a s ic p r o p e r t i e s of Hq:

1. E x a c tn e s s . If 0 ------ > • ---»- 0 is an exac t se quence ofsh e a v e s o v er M, we have an a s s o c ia te d long exac t se quence of cohom ologyg roups :

. . . . ------- H\эг2) - Í - * H4( ^ ) ----- H4+1( - U . НЧ+1(Щ)----- - ..........

2. H°(5<; = Г (M, (sec tions of OF).

3. N a tu ra l i ty . If f : Щ ------ »■ Щ is a sh e a f h o m o m o rp h ism , we have aninduced m a p on cohom ology g roups f:..:

Thus the m a p s j and i in the exac t cohom ology sequence a re induced f ro m the c o r re s p o n d in g m a p s in the o r ig in a l exac t se quence of she av e s .

F o r f u r th e r d e ta i l s and p roofs of the above c o n s t ru c t io n s we r e f e r to Refs [9, 14]. It is w orth r e m a r k i n g tha t we need p a r a c o m p a c tn e s s of M to p ro v e e x a c tn e s s of the long exac t se quence of cohom ology if we define sh e a f cohom ology v ia the Cech co n s t ru c t io n . It should be noted tha t p r o p e r ty 2 above gives an im m e d ia te function th e o re t ic d e s c r ip t io n of H° fo r sh eav es of functions. F o r exam ple , H°(M, & u ), the se t of ho lom orph ic functions on M.

2. STEIN MANIFOLDS

The f i r s t t r iu m p h a n t app l ica tions of sh e a f cohom olog ica l m ethods to p r o b le m s in com plex m an ifo ld s w e re the c e le b ra te d th e o re m s A and В of C a r ta n . We r e c a l l f i r s t of a l l the defin ition of a Stein m anifo ld :

(i) M is h o lom orph ic a l ly convex (for defin ition , s e e Ref. [8] , th e se P ro c e e d in g s ) .

(ii) A(M) s e p a r a t e s po in ts : Given a, b £ M , а ф b, 3 f A(M) such tha t f(a) ф f(b).

(iii) L o ca l c o - o r d in a te s on M can be defined by g lobally defined ana ly tic functions [8].

B e fo re s ta t in g th e o re m В we need to define what we m e a n by a co h e ren t sheaf . We say ^ is a c o h e re n t sh e a f (of 0M -m o d u le s ) on M if ^ has a r e s o lu t io n by f r e e sh e a v e s in so m e neighbourhood of ea ch point of M. That i s , if x e M , th e re e x i s t s a neighbourhood U of x and pos i t ive in te g e r s p and q such tha t the following sequence is exact:

04----- 0p----- ^ 1 u --- -0и и 1

F o r no ta t ions and te rm in o lo g y we r e f e r to Ref. [8] , th e se P ro c e e d in g s . !

1 Our definition is deceptive: The weight of coherence lies in properties of the Oka sheaf, 0 ^ , and Oka*s theorem . We refer to Ref. [18] for a fuller exposition.

170 FIELD

It follows f ro m th is defin ition and a fu n dam en ta l th e o re m of Oka tha t if------»-j3?2------► J83 is a sequence of c o h e ren t sh e av e s such that the

sequence of s ta lk s л/г x----- x ------------ * ^ 3 x is exac t at so m e point z 6 M, thenit is exac t in so m e neighbourhood of z. This p r o p e r ty is c h a r a c t e r i s t i c of co h e ren ce : " s ta te m e n ts o r p r o p e r t i e s about one s ta lk hold in a neighbourhood of the s ta lk " . T h is fac t enab le s one often to an a ly se p ro b le m s on a s ing le s ta lk (a lg e b ra ic a l ly ) and then get a s ta te m e n t holding in a neighbourhood of the s ta lk (" topologica l") .

T h is techn ique tu rn s out to be e x t re m e ly pow erfu l in app l ica tions . E x a m p le s of co h e ren t sh e a v e s a r e g iven by:0 M; the sh e a f of g e r m s of se c t io n s of any h o lom orph ic v e c to r bundle; the id e a l o r s t r u c t u r e sheaf of an ana ly tic se t . F o r p roo fs and m o r e d e ta i l s about c o h e re n c e we r e f e r to Refs [14, 18].

We m ay now s ta te th e o re m B.

T h e o re m В (H. C artan ) :

If c®' is a co h e ren t sh e a f on a S tein m anifo ld M, then H4(M,_rf] = 0, q â 1.

Let u s f i r s t m a k e one o r two o b se rv a t io n s about th is r e s u l t . F i r s t l y i t is an ex a m p le of a cohom ology van ish in g th e o re m . That is, a l l s t r i c t ly p os i t ive d im e n s io n a l c o h e re n t sh e a f cohom ology v a n ish e s . T h is is c h a r a c t e r i s t i c of the type of r e s u l t tha t is in te r e s t in g in the th e o ry (It is say ing som e th ing about a lack of o b s t ru c t io n s to so lv ing v a r io u s types of p ro b lem involving the sheaf). M o re g e n e ra l ly , we m igh t have r e q u i r e d so m e type of f in i te n ess ; fo r exam ple , d im НЧ(У) < » .

Next we o b se rv e tha t the v an ish ing of c o h e re n t sh e a f cohom ology im p l ie s , often v e r y e a s i ly , s t ro n g r e s u l t s about the s t r u c t u r e on the o r ig in a l m anifo ld . To be p r e c i s e , we sh a l l p rove p a r t of:

P ro p o s i t io n :

L et M be a com plex m an ifo ld such tha t H4 (M,j?/) = 0, qS 1, fo r al l c o h e re n t sh e a v e s jé on M. Then M is Stein.

(In fac t , we sh a l l only need to know tha t H1 (M, ji) = 0 fo r a l l co h e ren t sh e a v e s of id e a ls on M. )

P roo f :

The p ro o f of th is r e s u l t is ea sy , the m o r e so by c o m p a r is o n with the p ro o f of T h e o re m B, and p ro v id es a good exam ple of the advan tages of the s h e a f cohom olog ica l se tt ing .

Let u s show tha t M is h o lo m o rp h ic a l ly convex. Suppose tha t К is any com pac t su b se t of M. Suppose К is not com pac t. Choose an infinite s u b se t Q of К with no l im i t po in t in M. T h en Q is an a la ly t ic s u b se t of M with an a s s o c ia te d id e a l sh e a f ¿d, say . (R ecal l tha t is the sh e a f of g e r m s of ana ly tic functions on M van ish ing on Q. ) Then is a co h e ren t sh e a f of id e a ls and so Н 1(М,^г/) = 0.

IAEA-SM R-11/23 171

A sso c ia te d to the exac t sh e a f se quence 0 ------» - j / ------»-<9M----- Ы -------------•’Owe have the long exac t sequence of cohomology, the f i r s t few t e r m s of which a re :

0 ------► Н ° Ы ) ----- - H°( & ) ------► B°(0 ¡já)----- ► h V ) ------►.........M M

Hence we obta in the s h o r t exac t sequence:

0 — ► H 'V ) — ► h V m ) — - h V m / * o — ►o

Now, H°(0M/jsO = T{0 j^ ) is ju s t the s e t of ana ly tic functions on Q. But s in c e Q is an iso la te d se t of po in ts a function on Q is uniquely d e te rm in e d by a s s ig n in g an a r b i t r a r y com plex n u m b e r to ea ch point of Q. SinceH °(0M) ------«• H °(¿?M/ V ) ------*■ 0, it follows tha t ev e ry ana ly tic function on Q isthe r e s t r i c t i o n of an ana ly tic function on M. T hat is , we m ay choose an e lem e n t of A(M), say f, taking an a r b i t r a r i l y a s s ig n e d s e t of va lues on Q. In p a r t i c u la r , we m ay a s su m e tha t f is not bounded on Q and hence on К and th e re fo r e , by defin i t ion of K , on K. C on trad ic t ion , s ince К is com pact.

The o th e r conditions fo r a Stein m anifo ld m ay be equally t r iv ia l ly checked to be t ru e . F o r the d e ta i l s , we r e f e r to a s e m in a r by S e r r e in Ref. [29].

Thus f a r we have se e n that knowledge about the cohomology g roups of s t r u c t u r e s can im ply s t ro n g r e s u l t s about the s t r u c t u r e s and the under ly ing topo log ica l space . H owever, again c h a r a c t e r i s t i c a l ly , the p roo f of cohom ology van ish ing in T h e o re m В is ha rd . F o r a p ro o f involving the th e o ry of e l l ip t ic p a r t i a l d i f fe re n t ia l equations on n o n -co m p a c t m an ifo lds (proving the 9 sequence exact) to g e th e r with a n o n - t r iv ia l ap p rox im ation a rg u m e n t due to C a r t a n we r e f e r to Ref. [18] . F o r a p roo f not involving e l l ip t ic th e o ry we r e f e r to Ref. [14] o r Ref. [29].

T h e o re m В is of p a r t i c u la r im p o r ta n c e as Stein m an ifo lds prov ide the lo c a l m o d e ls fo r com plex m anifo lds and the cohomology van ish ing gives a s t r a ig h t fo rw a rd technique fo r com puting sh e a f cohomology for a r b i t r a r y com plex m an ifo lds . To be p re c i s e : le t ^ " b e a c o h e ren t sheaf on the com plex m anifo ld M and le t <&'' = {U¡} be an open co v e r of M by Stein manifo lds .M aking the o b se rv a t io n tha t an in te r s e c t io n of Stein m an ifo lds is s t i l l Stein, th e o re m В im p l ie s tha t Hq(Us, ¿F) = 0, q§ 1, w here Us = U¡ Л . . . n u ¡ . The th e o re m of L e ra y (a d i r e c t a rg u m e n t o r e le m e n ta ry applica tion of s p e c t r a l sequences) then gives

Hq( « / ,^ ) s H q(M ,á n . (For d e ta i l s , s e e Ref. [9] )

An analogous s i tua t ion m ay be shown to hold in the study of ana ly tic s e ts and a lg e b ra ic g e o m e try w h ere one has lo c a l m ode ls (Stein ana ly tic sp a ce s and affine sc h e m a ta , re sp e c t iv e ly ) which again m ay be p roved to have van ish ing sh e a f cohomology.

B efore tu rn in g to o the r ex am ples of van ish ing th e o re m s le t us note one fu r th e r app l ica t ion of th e o re m В to a p ro b lem in m e ro m o rp h ic functions, nam ely the C ousin I p ro b lem .

172 FIELD

With the no ta tion of Ref. [8 ] , le t be the sh e a f of g e r m s of m e ro m o rp h ic functions on the Stein m anifo ld M is not a co h e ren t sheaf).

L et {Uj} be a c o v e r of M and suppose we a r e g iven the p r in c ip a l p a r t of a m e ro m o rp h ic function on ea ch U¡ , m¡G ( U j ) , such tha t mj-mj-G A ( U ¡ n U j ). T hen th e re e x i s t s a m e ro m o rp h ic function m , defined on the whole of M, such tha t m - m ¡ G A ( U ¡ ) . (This th e o re m is the s e v e r a l com plex v a r ia b le ana logue of the M it ta g - L e f f l e r th e o re m [18]).

The p ro o f is t r iv ia l . We have the s h o r t exac t sequence

which y ie ld s , to g e th e r with th e o re m B, the following p o r t ion of the long exac t se quence of cohomology:

H°( 0 н ° с ^ ■ h u( . ^ M

A(M)

But the m¡ define a se c t io n ф of 0M ( r e c a l l m ¡- mj G A(U¡ П Uj )). Hence, th e re ex is ts m G Г wi t h p(m) = 4. C le a r ly ,m - m ; G A(U¡).

3. THE LEVI P R O B LEM AND VANISHING OF COHOMOLOGY

L et us f i r s t of a l l define a (s trong ly ) p seu d o -c o n v ex m anifold . Thus, suppose M is contained in the com plex m anifo ld M as a r e la t iv e ly com pac t

Mopen com plex subm anifo ld .M is sa id to be s t ro n g ly pseudo- convex if th e re e x i s t s a d if fe ren t ia b le(say C2) function ф : M ------*- R such that:

(i) M = {xGM: ф (x)< 0}.(ii) Dф ¡ Э М ^ О (This im p l ie s 3M

is smooth).(iii) T he L e v i - f o r m of Ф,

ддф G С” (T*M ® Т*М) is pos i t ive defin itewhen r e s t r i c t e d to 3M (Locally ддф is the H e r m i t ia n m a t r ix given by d24 / d z ldzj ).

T he o r ig in a l e x a m p le s of s t ro n g ly p seu d o -c o n v ex m an ifo lds o c c u r re d in p r o b le m s connected with dom ains of ho lom orphy (here M = (Em) when Levi o b se rv e d tha t do m a in s of ho lom orphy with su ff ic ien tly sm oo th b o u n d a r ie s a r e p se u d o -c o n v ex and co n jec tu re d the co n v e r se to be t r u e . T h is co n je c tu re w as p roved f i r s t by Oka fo r dom ains in Œ2 and l a t e r fo r the g e n e r a l c a s e by B r e m e r m a n n , N orgue t and Oka independently R efs [5, 26, 27 ] . The g e n e r a l p ro b le m of the study of convexity conditions on the boundary of a dom ain and th e i r r e la t io n to th e function th e o ry of the dom ain is of g r e a t s ign i f icance in

IAEA-SM R-11 /2 3 173

the a n a ly s is of b oundary va lue p ro b le m s in p a r t i a l d i f fe re n t ia l equa tions . We r e f e r to R e f s [23] , [24], fo r d i s c u s s io n s of the com plex p seu d o -c o n v ex c a se and to Ref. [19] fo r a m o r e g e n e r a l d i s c u s s io n val id fo r the r e a l c a se .

The g e n e ra l iz e d L ev i p ro b le m is to p ro v e th a t a s t ro n g ly p seu d o -c o n v ex m anifo ld is ho lo m o rp h ic a l ly convex. G ra u e r t in 1958 Ref. [10] p roved tha t if M is s t ro n g ly p seu d o -c o n v ex then НЧ( М , ^ ) is f in i te -d im e n s io n a l fo r q> 0 and & a co h e ren t sheaf . U sing th is r e s u l t he then p ro v ed th a t M is ho lom o rp h ic a l ly convex us ing a s t r a ig h t f o rw a r d a rg u m e n t of the type u se d to p ro v e the c o n v e r s e to th e o re m B. H ow ever, the cohom ology does not v an ish in g e n e r a l and so M is not Stein. A r e a s o n a b le ques tion is to a sk how f a r away f ro m a Stein m an ifo ld is a p se u d o -c o n v ex m an ifo ld and to p r e c i s e ly c h a r a c te r i z e the o b s t ru c t io n s to its be ing Stein. We sh a l l b r ie f ly re v ie w so m e w ork of R o s s i (see Ref. [28]; th is p a p e r a l so con ta ins an ex tens ive b ib l io g raphy on the L ev i p ro b lem ) which a n s w e rs th e se ques t ions in a cohom olog ica l f ra m e w o rk .

The f i r s t point to note about a n o n -c o m p a c t com plex m anifo ld M is tha t if it ha s any com pac t s u b v a r ie t ie s it cannot be Stein as we c l e a r ly cannot s e p a r a te po in ts of M lying on the s a m e connected com ponen t of a p o s i t iv e d im e n s io n a l co m p ac t su b v a r ie ty . In the c a s e w h ere M is s t ro n g ly pseudo- convex it m a y be shown tha t the o b s t ru c t io n to M being Stein is r e p r e s e n te d by the p o s i t iv e -d im e n s io n a l com pac t ana ly tic su b sp a c e s of M. The p r e c i s e r e s u l t is:

T h e o re m :

If M C M is s t ro n ly p se u d o -convex , then(a) If x and у belong to d if fe ren t connected s u b v a r ie t i e s of M, th e re ex is ts

an f G A(M) such tha t f(x) ф f(y).(b) If x does not belong to any s t r i c t l y p o s i t iv e -d im e n s io n a l co m p ac t sub-

v a r ie ty of M, th e re ex is t f j , . . , f n e A(M) which give lo c a l c o - o r d in a te s at x.

(c) M has only f in ite ly m any p o s i t iv e - d im e n s io n a l co m p ac t s u b v a r ie t ie s .

T h is th e o re m is deduced f ro m the following cohom olog ica l r e s u l t which g ives in fo rm a tio n about the su p p o r t of cohomology:

T h e o re m :

If M C M is s t ro n g ly p se u d o -c o n v ex and J ' ' is a c o h e re n t sh e a f on M, then fo r p>0 th e r e e x is t f¡ G A(M), 1 S i s t such that(a) If I deno tes the id e a l sh e a f g e n e ra te d by f j , . . , f t then the r e s t r i c t i o n m ap

Hp( M ,3 r ) ------- H P(M, ©yjI ® j r ) is in jec t ive .M

(b) V = {x e M:fj (x) = 0 I s i s t } c o n s is t s of f in i te ly m a n y le v e l s e t s . ( A l e v e l s e t is of the fo rm : L c = {z e M :f (z) = с fo r a l l f e A(M)} fo r so m e c€(C. )

T h is th e o re m has the obvious c o r o l l a r y tha t if M has no com pac t p osit ive d im e n s io n a l s u b v a r ie t ie s then M is Stein. The m a in e lem e n t in the p roo f of th is th e o re m is the o b se rv a t io n tha t the p seudo-convex iv i ty condition fo rc e s the co m pac t s u b v a r ie t ie s of M to be bounded away f ro m the boundary of M.

174 FIELD

F o r m o r e d e ta i l s we r e f e r to Ref. [28]. It ought a l so to be r e m a r k e d tha t the u se of p a r t i a l d i f fe re n t ia l equation techn iques al low s of the p o ss ib i l i ty of g iv ing p r e c i s e e s t im a te s on the g row th of functions on M at the boundary , a f e a tu re tha t is una t ta inab le by p u re ly sh e a f th e o re t ic techn iques . We a lso r e m a r k tha t by identifying ea ch connected co m p ac t su b v a r ie ty of M to a point we find tha t a s t ro n g ly p seu d o -c o n v ex m anifo ld M is a p r o p e r m odif ica tion of a Stein ana ly tic sp a c e with iso la te d s in g u la r po in ts . C o n v e rs e ly , by H iro n a k a 's r e so lu t io n of s in g u la r i t ie s , a Stein ana ly tic sp a ce with iso la te d s in g u la r i t i e s m ay be blown up into a p se u d o -c o n v ex m anifo ld . F o r m o r e d e ta i l s and r e f e r e n c e s fo r app l ica tions we r e f e r to Ref. [28] .

The above d e s c r ip t io n of a s t ro n g ly p seu d o -c o n v ex m anifo ld was a r e l a t ive one depending as i t did on the sp a ce M. We w ill now look at a s l igh tly d if fe ren t defin ition of p seudo-convex iv i ty which has the advantage of being in t r in s i c , though we lo se the connect ions with boundary va lue p ro b le m s in p a r t i a l d i f fe re n t ia l equa tions .

F i r s t , by way of m o tiva t ion , we r e m a r k tha t if we have a dom ain of ho lom orphy С (En and define the function p(z) = - lo g (d(z, 3Í2 )), then the L ev i f o rm of p, ЭЭр, is a pos i t ive defin ite H e rm it ia n fo rm . (At le a s t , if the b oundary is sm oo th enough o r a l te rn a t iv e ly we in te r p r e t th is s ta te m e n t d i s t r ib u t io n a l ly .) T h is su g g e s ts the following defin ition: Suppose M is a co m plex m anifo ld and th e re e x i s ts a function ф : M ------»■ R such tha t :

(i) Mc = { z€M : ф S c} is com pac t fo r a l l c e R.(ii) L(</))= ддф has n -p p os i t ive e igenva lues at ea ch point in M.

Then we c a l l M a s t ro n g ly p -p se u d o -c o n v e x o r p -c o m p le te m anifold .T h is defin ition m a y a lso be fo rm u la te d fo r ana ly tic s p a c e s , but we

r e s t r i c t a t ten t ion h e r e to m an ifo lds and r e f e r the in te r e s te d r e a d e r to Refs [1 ,2] fo r the ana ly tic sp a ce c a se . To avoid confusion with p re c e d in g m a t e r i a l on the L ev i p ro b lem we sh a l l u s e the t e r m p -c o m p le te r a t h e r than p -p seu d o -co n v e x .

A dom ain of ho lom orphy is thus an exam ple of a 0 - com ple te m anifold (proof in Ref. [18] ).

T h e o re m В has g e n e ra l iz a t io n s t o p - c o m p l e t e m an ifo ld s . F i r s t we r e c a l l tha t a ho lo m o rp h ic p - f o r m Ф on M is ju s t a d i f fe re n t ia l fo rm of type (p ,0) such th a t Ф is h o lom orph ic (and so Ъф = 0). If E is a ho lom orph ic v e c to r bundle on M then we le t f ip (E) denote the s e t of ho lom orph ic p - f o r m s on M with va lu es in E . We w ill a l so le t f2p(E) denote the sh e a f of g e r m s of holom o rp h ic E va lued p - f o r m s on M.

We have the following v an ish ing th e o re m [3, 31] :

T h eo rem :

If M is p -c o m p le te , then:(i) H s(M,f2r (E ) ) = 0, s s p + l , r ê 0

(ii) H*(M,f2r (E)) = 0, s S n - p - 1. "H*" deno tes cohom ology with com pac t s u p p o r ts .

IA EA -SM R-11/23 175

The p ro o f is a g e n e ra l iz a t io n of the p a r t i a l d i f fe re n t ia l equation th e o r e t i c p ro o f of the e x a c tn e s s of the 9 - sequence .

T h is type of r e s u l t can be of u s e , fo r ex a m p le , when we re m o v e a c o m p lex subm anifo ld of a co m p ac t com plex m an ifo ld and a r e able to p ro v e tha t the n o n -c o m p a c t m an ifo ld r e s u l t in g is p -c o m p le te . We w ill e n c o u n te r such an app l ica t ion la te r .

4. VANISHING THEORY ON COMPACT C O M P L E X MANIFOLDS:EMPHASIS ALGEBRAIC

In s e c t io n s 2 and 3 we ind ica ted tha t v an ish ing of c o h e re n t sh e a f cohom ology im p l ied s t ro n g r e s u l t s about the e x is te n c e of ana ly tic and m e ro m o rp h ic functions on the u nder ly ing com plex m anifo ld . T he f i r s t th e o re m we have fo r co m p ac t com plex m an ifo lds is the f in i te n e s s th e o re m of C a r t a n - S e r r e :If .S 'is a c o h e re n t ana ly tic sh e a f on the co m p ac t com plex m an ifo ld M, then H ^ M , ^ -) is f in i te -d im e n s io n a l fo r qêO ,Refs [7 ,1 4 ] . Thus if & i s the sh e a f of s e c t io n s of a h o lom orph ic v e c to r bundle the th e o re m sa y s tha t isf in i te -d im e n s io n a l . T hat i s , the s e t of s e c t io n s of a ho lo m o rp h ic v e c to r bundle o v e r a com pac t com plex m anifo ld n e c e s s a r i l y f o rm s a f in ite d im e n s io n a l v e c to r sp a ce . (It is an im p o r ta n t p ro b le m to be able to com pute the d im e n s io n of th is v e c to r sp a c e in t e r m s of topo log ica l in v a r ia n ts of the bundle and of M : The m o s t im p o r ta n t th e o re m s c o v e r in g th is type of s i tua tion a r e the th e o r e m s of the G ro th e n d ie c k -H irz e b ru c h -R ie m a n n -R o c h type to g e th e r with th e i r g e n e ra l i s a t io n to com plex and d if fe ren t ia b le manifolds^the- A tiyah -S inge r index th e o re m . We r e f e r the r e a d e r to Ref. [16] , and a lso to Ref. [4] fo r d e ta i l s and app l ica t ions of the index th e o re m . )

Unfortunately , the th e o re m of C a r t a n - S e r r e , though v e r y u se fu l in c e r t a in c a s e s , does not r e a l ly p ro v id e m u c h new in fo rm a tio n about the sh e a f yr o r M. H ow ever, in the c a s e of a lg e b ra ic m a n ifo ld s , Ref. [8 ] , we a r e able to p ro v e a pow erfu l van ish in g th e o re m , due to S e r r e , which is analogous to T h e o r e m В fo r S tein m an ifo lds and c h a r a c t e r i s e s a lg e b ra ic m an ifo ld s . The p ro b le m of s tudying the c a s e when M is not a lg e b ra ic and finding o b s t ru c t io n s to M being a lg e b ra ic o r belonging to so m e o th e r "good" c l a s s of m an ifo ld s is s t i l l open. It is not un like ly tha t a s u c c e s s fu l study of m o r e g e n e r a l com plex m an ifo lds w ill depend heav ily on a g loba l d e fo rm a t io n th e o ry of com plex s t r u c t u r e s . In any c a s e we w ill not c o n c e rn o u r s e lv e s with th e se p ro b le m s h e r e but in s tea d r e s t r i c t o u r s e lv e s fo r the r e s t of th is s e m in a r to m an ifo ld s of an e s s e n t i a l ly a lg e b ra ic type.

To m o tiv a te the techn iques we sh a l l u se ,w e r e m a r k tha t the van ish ing of sh e a f cohom ology on a Stein m an ifo ld r e s u l t e d to som e ex ten t f ro m the ex is te n c e of p len ty of ana ly tic functions on M. Now on P n, w hils t th e re can be no n o n - t r i v ia l ana ly tic functions , n e v e r th e le s s , as we sh a l l s h o r t ly see . th e re a r e p len ty of m e ro m o rp h ic functions and the fo rm of th e o re m В for a lg e b ra ic m an ifo ld s w ill r e f l e c t th is fac t.

L et u s f i r s t of a l l show how m e ro m o rp h ic functions on com plex m an ifo lds c o r r e s p o n d to s e c t io n s of h o lom orph ic line bund le s . We m u s t f i r s t r e v ie w the defin ition and p r o p e r t i e s of d iv is o rs .

L et ©''' be the sh e a f of g e r m s of in v e r t ib le ana ly tic functions on M.0 * С 0 and 0 * is ju s t the su b se t of 0X of g e r m s of functions which do not v an ish a t x. We have s u p p r e s s e d the M in 0 M in the above: we sh a l l continue to do th is u n le s s th e r e is d a n g e r of confusion.

176 FIELD

L e t . ^ ’' be the sh e a f of g e r m s of in v e r t ib le m e ro m o rp h ic functions on M. H ere and „ ^ ,s = . . ^ - {0} . N e i th e r 0 * nor-.-^* a r e cohe ren t .

Le t */£?*. eg) is ca l le d the sh e a f of g e r m s of d iv is o rs on M. Ase c t io n of Ü? is ca l le d a d iv iso r .

If d G Г ( ÇZ>) and {U¡} is a co v e r of M, then d is g iven by a co l lec t ion { dj}, d iG . ^ * ( U i ) ,such tha t d jd ^ G 0 * (U ^ n U j ).

N otice tha t if d is a d iv is o r then it def ines an ana ly tic su b se t D of M.In fac t , D П U¡ is defined to be the union of the pole and z e r o s e t of d¡.

A sso c ia te d to d we a lso have a line bundle [d] whose t r a n s i t io n functions a r e g iven by g¡j = djd^1. A nother way of se e in g th is r e la t io n is to note tha t ~ft\M,0*) is iso m o rp h ic to the g roup of ho lom orph ic l ine bundles on M (group o p e ra t io n t e n s o r p roduct) . Then we note th a t we have the cohom ology exactse q uence a s s o c ia te d to 0 ------► 0 * ------¿f*----------*■&)----- ► 0, giving u s these q uence Г (*-<?*)------*■ Г { & ) ------*• H ^M , 0* )------«-................ In th is sequence ,d e r ( ^ ) goes t o - [ d j E H ^ M . i ? * ) .

We say tha t a d iv is o r is p r in c ip a l , if it is the im age of a s e c t io n of ..4?'.It then follows tha t it is defined by a g loba l m e ro m o rp h ic function and the c o r re sp o n d in g l ine bundle is then h o lo m o rp h ic a l ly t r iv ia l .

Suppose d is a d iv i s o r and F = [d] is the a s s o c ia te d line bundle . If d is g iven loca l ly on the c o v e r {U¿} by d ¡ , then F has t r a n s i t io n functions gjj = ¿¡d"1 and so F k= ® kF has t r a n s i t io n functions g¡j = d ¡d jk (к m a y be p o s i tive o r negative).

Suppose tha t Ф e F (M , F k). L oca lly we have ф1 = gk <#>j on Uj Л U j . Hence, s in c e g.k = dHdJk, follows tha t <í>¡/dk = <í>j/dk. In o th e r w o rd s , th e re ex is ts a m e ro m o rp h ic function m on M such tha t m = «¿’¡ /d! ' on Uj. Thus s e c t io n s of F k c o r r e s p o n d in a n a tu r a l way to m e ro m o rp h ic functions on M.

If djGACUj) (that i s , d¡ has no po le s ) , i t is a l so c l e a r tha t m | Uj has po les only on the z e r o s e t of dj ; tha t i s ,m only has po les on D.

It is not, h ow ever , a p r i o r i c l e a r tha t a g iven l ine bundle has any nont r i v i a l se c t io n s : take fo r in s tan c e the t r i v i a l bundle.' In fac t , m any line bund les w il l have no se c t io n s at a l l , o th e r than the z e r o sec tion . One of the s p e c ia l f e a tu re s of a lg e b ra ic m an ifo ld s is th a t th e re e x is t l ine bundles with p len ty of se c t io n s .

We now in tro d u ce a defin ition th a t w ill enab le us to s ta te an im p o r ta n t van ish in g th e o re m .

Definition:

Suppose V is a s u b se t of the com plex m an ifo ld M. We say V can be blown down to a point if th e r e e x i s ts an ana ly tic sp a c e X, point x £ X and am a p f : M ------»-X such that:

(i) f(V) = x(ii) f : M - V ------»-X-{x} is an ana ly tic iso m o rp h is m .T h is def in i t ion l inks our w ork on com plex m an ifo lds with tha t on s t ro n g ly

p se u d o -c o n v ex m an ifo lds fo r , if V can be blown down to a point x, i t follows (by looking at in v e r s e im a g es of s t ro n g ly p se u d o -c o n v ex neighbourhoods of x) tha t V has a fu n d am e n ta l s y s t e m of s t ro n g ly p seu d o -c o n v ex neighbourhoods.

In the next se c t io n , we s h a l l be ta lk ing m a in ly about notions of pos i t iv i ty fo r a v e c to r bundle . H ere we in tro d u ce the w eakes t defin ition of posit iv i ty , due to G ra u e r t , Ref. [11] .

IA EA-SM R-11/23 177

If F ------»- M is a h o lom orph ic v e c to r bundle we say:(i) F is w eakly n eg a tiv e , if the z e ro se c tio n of F can be blown down to

a point.(ii) F is w eakly p o s itiv e if F* is w eakly n eg a tiv e .T he m o s t im p o rta n t exam ple of a w eakly p o s itiv e line bundle is the hyp erp lan e

se c tio n bundle of P n. A sso c ia ted to the h y p erp lan e H = {( z j , . . , zn + 1) : z 2 = 0} we have a d iv is o r defined by zj = 0. If [H] d eno tes the co rre sp o n d in g line bundle it m ay be checked th a t the tr a n s i t io n functions of [H] a r e given by gy = Z j/Z j. H ere we have tak en the can o n ica l open co v e rin g of P n by n + 1 open su b se ts . To show th a t [H] is w eakly p o s itiv e we p ro v e th a t [H] is w eakly n eg a tiv e . But [H] ' m ay e a s ily be show n to be the can o n ica l linebundle on P n induced fro m the n a tu ra l p r in c ip a l bundle on P n : (En+1- {0}----► P n.T hen we ju s t no te th a t th e re is a n a tu ra l m ap ф : [H] * - 0 - s e c tio n -> (En+1 -{0} w hich is an iso m o rp h ism and as an im m ed ia te co n seq u en ce the z e ro -s e c t io n of [H] * m ay be blow n down to a point.

If V С P n is a subm an ifo ld no te th a t [H]|V is w eakly p o s itiv e . Thus e v e ry a lg e b ra ic m an ifo ld ad m its a w eakly p o s itiv e lin e bundle .

We have the follow ing w eak fo rm of the K o d a ira v a n ish in g th e o re m :

T h eo rem :

L e t M be a co m p ac t com plex m an ifo ld and F a w eakly p o s itiv e line bundle on M. Thenîf 7 is a c o h e re n t sh e a f on M, th e re e x is ts k(7 , F) G Z+ su c h th a t fo r a l l k ê k(7 , F):

НЧ(М, 7 ® F k) = 0 fo r q ê 1

Definition:

The con ten t of the th e o re m is th a t, even if Hq(M ,7 ) does not v an ish , we ca n tw is t the sh e a f 7 by te n so r in g w ith F k to get v an ish in g of cohom ology.

The p ro o f of th is r e s u l t m a k es e s s e n t ia l u se of the fac t th a t if D is a s tro n g ly p seu d o -co n v ex neighbourhood of the z e ro se c tio n of F * , then Hq(D, i^) is f in ite d im e n s io n a l fo r q = 1 and & a c o h e re n t sh e a f on F*.

We lif t the sh e a f 7 on M to the sh e af

7 = tt*7 ®- 0 * on F*M

(again co h e ren t) and show th a t th e re e x is ts a n a tu ra l m ap:

N

j : ^ Hq(M , 7 ® F k ) ------► H q(D3y)k = 0

w hich m ay be p roved to be an in jec tio n . M aking u se of the fin ite d im ens io n a lity of H q(D, 3f ) , fo r J*" a c o h e ren t sh e af on F * , the re s u l t follow s at once. F o r fu r th e r d e ta ils we r e f e r to Bef. [7] .

B efo re s ta tin g the m a in ap p lica tio n of the above th e o re m we give a defin ition :

178 FIELD

If F is a ho lom orph ic line bundle & such tha t fo r a l l c o h e re n t sh e a v e s & on M, th e re e x i s t s k (F , &) G Z+ such that:

H4(M, 3?® F k) = 0, fo r a l l k È k ( 3 s F ) and q ê 1

then we ca l l the l ine bundle F , cohom olog ica lly posit ive .

Definition:

S e r r e p roved tha t the hy p erp la n e sec t io n bundle of P n d e s c r ib e d above is cohom olog ica lly posit ive ,R ef. [29]. In fac t , he p roved that:

Hq( P n, 0 ® [H] n ) = 0, q ê 1, пй 0. (F o r p roofs s e e Ref. [29]. )

The im p o r ta n c e of the notion of cohom ologically pos it ive follows f ro m the fundam en ta l r e s u l t of S e r r e which g e n e r a l iz e s th e o re m s A and В for Stein m an ifo lds :

T h e o re m :

M is a lg e b ra ic if and only if th e re e x i s ts a cohom ogica lly pos it ive l ine bundle on M.

The im p l ica t io n tha t cohom olog ica l p os i t iv i ty im p l ie s M is a lg e b ra ic is not d ifficu lt and in s p i r i t follows tha t of the c o n v e r se to T h e o re m В fo r Stein m an ifo ld s :

Let us f i r s t show tha t the condition of cohom olog ica l pos it iv i ty fo r a line bundle F o v er M im p l ie s tha t th e re ex is t p lenty of s e c t io n s of F k fo r su ff ic ien t ly la rg e k. Suppose, th e r e f o r e , tha t a G M. We show that th e re ex is ts к G Z , such tha t th e re ex is ts a s e c t io n ф of F k with ф(a) =/= 0. If .У denotes the id e a l sh e a f of the ana ly tic se t a and Ф = 0 / j ? , we have the exac t sequence:

0 ----- »■ j ? ----- - 0 ----- - V ----- ► 0

C le a r ly , the sh e a f Ф is i so m o rp h ic the sheaf Ca whose s ta lk is z e r o , except at the point a w here it is i so m o rp h ic to Œ.

T e n s o r in g with F k, fo r su ff ic ien tly la rg e k, and u s in g the exac t cohom ology se quence we obta in the sequence :

r ( F k) ----- * F (F ak) ------» 0

s ince F k ) = 0. T hus , given an a r b i t r a r y point zG Fak , th e re ex is tsa sec t ion r of F k such that т (a) = z.

Let s 0 , . . . , sk be a b a s i s of se c t io n s of F N fo r suffic ien tly la rg e N.Then we have a m a p M ----- « - P k which is defined loca l ly by m appingxl------*■ (s0( x ) , . . . , s k(x)). It m ay e a s i ly be checked tha t th is m ap is independent of lo c a l r e p r e s e n ta t io n for the s t u sed . The p ro b lem is then to choose

IAEA-SM R-11 /23 179

N so tha t th is m ap is an em bedding. T h is is done by e x p r e s s in g the p ro b lem in t e r m s of su i tab le c o h e re n t sh e av e s and u s in g the cohom olog ica l positivity. F o r m o r e d e ta i l s we r e f e r to Ref. [7] .

Let us now m ake a s im p le app l ica tion of the ex is te n c e of a cohom o- log ica l ly pos it ive line bundle on P n to p rove C how 's the o re m , Ref. [8].

Suppose H is the hy p erp la n e s e c t io n bundle a s s o c ia te d to the d iv iso r Í z-l = 0 } . L e t u G Г (P n, [H]k ), k ï O .

We have the following

L em m a:

Any se c t io n of [H]k is n a tu ra l ly iden tif ied with a hom ogeneous po lynom ial of d e g re e к in the co o rd in a te s (Zj, . . . , z n+1 ).

The p ro o f follows ea s i ly . Let Uj = {(z j , . . , z n + 1 ) : z¡ ф 0} , th e n U j = € n . Suppose p is a hom ogeneous po lynom ial of d e g re e к on <Tn + 1. p defines a ho lo m o rp h ic function f ^ p / z ^ o n UjC P n fo r ea ch i, s ince p /z 'j is hom ogeneous of d e g re e 0. Now fj = (z¡/ z j ) k f - h e n c e {f¡} def ines a s e c t io n of [ H Jk . Con- v e r s e ly , by r e v e r s i n g the a rg u m e n t , e v e ry s e c t io n of [H ]k is obtained in th is way.

We m a y now e a s i ly p ro v e the following th e o re m of Chow:

T h e o re m :

Let A be an ana ly tic su b se t of P n . T hen th e re e x is t hom ogeneous po lynom ials f 1; . . . , f m such tha t A is the com m on z e ro locus of the f ¡ .

P roo f :

We w ill p rove the following: L et b e P n-A . T hen th e re ex is ts a hom ogeneous po lynom ial v an ish ing on A such tha t f(b) =/= 0. Once th is is p roved we c o n s id e r the s e t of a l l hom ogeneous po lynom ials which van ish on A. Applying the H ilb e r t b a s i s th e o re m we choose a fin ite s u b se t of th e se po lyn o m ia ls whose com m on z e r o locus is p r e c i s e ly A.

Suppose then tha t deno tes the id e a l sh e a f of b and tha t F deno tes the sh e a f of s e c t io n s of the hy p erp la n e s e c t io n bundle of P n .

We have the exac t sh e a f sequence

0 ------ ► ^ ® ^ ® F m ------- J f ® F m------►j? ®F™ ----- ►OA b — A — A, b —b

F o r m la rg e enough, the cohom ology of J ^ ® _ ! £ ® F m in d im ens ion g r e a t e r than o r equa l to one v an ish es and so we have the following po r t io n of the exac t se quence of cohomology:

Г (P n, J?A ® F m ) ---------------- ► Г (P n, J?A_ b® Ffcm ) ------► 0

Hence th e r e e x i s t s f G Г ( Р П, j^® F m ) such tha t f(b)=^0. But s ince С €?n, it follows tha t Г ( Р П, J ^ ® F m ) С r ( P n,F_m). C le a r ly , Г ( Р П, УА® F m)

180 FIELD

is ju s t the su b sp a c e of r ( P n, F m) v an ish ing on A. Since , th e r e f o r e , f m ay be r e g a r d e d as an e lem e n t o f F ( P n, F m ) it follows by the above le m m a tha t f is the hom ogeneous po lynom ial whose ex is te n c e we a s s e r t e d .

In conclus ion : In th is se c t io n , we have given ano ther i l lu s t r a t io n of how knowledge of the sh e a f cohom ology of a com plex m anifo ld g ives im p o r ta n t in fo rm a tio n about the com plex s t r u c t u r e of the m anifo ld , th is t im e of an a lg e b ra ic n a tu re .

5. POSITIVITY OF VECTOK BUNDLES AND VANISHING THEORY ON COMPACT COM PLEX MANIFOLDS: EMPHASIS D IF F E R E N T IA L GEOMETRIC

In the r e m a in d e r of th is s e m in a r I w ish to s u rv e y som e w ork of G ri f f i th s , g e n e r a l i s in g w ork of K o d a ira , on v a r io u s notions of pos it iv i ty fo r v e c to r bund les . The m a in r e f e r e n c e w ill be Ref. [12] , though H a r t s h o r n e 's paper. Ref. [15], should a l so be consu lted fo r an a lg e b r o - g e o m e tr ic p re se n ta t io n .

The defin ition of a weakly pos i t ive bundle g iven in the p rev ious sec t io n does not s e e m to allow of p roofs of p r e c i s e v an ish ing th e o re m s . H ere , we sh a l l in tro d u ce th r e e new defin itions of p os i t iv i ty fo r v e c to r bund le s , one of a d i f fe re n t ia l g e o m e tr ic n a tu re which a l lows one to p ro v e p r e c i s e van ish ing th e o r e m s , one of a com plex ana ly tic n a tu re which e x p r e s s e s the fac t tha t the bundle has 'p len ty ' of ana ly tic se c t io n s and f inally a defin ition of pos it iv i ty of a topo log ica l n a tu re involving C h e rn c l a s s e s .

Our defin i t ions will follow the p ap e r of G riff i th s [12] r a t h e r than tha t of H a r t s h o r n e , though with one o r two m in o r d i f fe ren ce s of de ta i l .

In the p re c e d in g se c t io n , we only defined cohom olog ica lly pos it ive line bund les . In g e n e ra l iz a t io n s to v e c to r bundles we work, fo r te ch n ic a l r e a s o n s , with s y m m e t r i c t e n s o r p o w ers of the bundle:

Definition:

The h o lom orph ic v e c to r bundle E —— »- M is sa id to be cohom ologically po s i t iv e if, fo r e v e ry co h e ren t sh e a f S'- on M, th e re ex is ts k(E, ¡P) £ Z + such that:

H 4(M, . ÿ '® E (k)) = 0, qS 1 and k ï k (E,á?)

fk)H ere E l deno tes the к - th s y m m e t r i c t e n s o r pow er of the bundle E , s o m e t im e s denoted by OÊ.

R e m a rk :

U sing an a rg u m e n t analogous to tha t ske tched in the p rev io u s sec t ion , w h ere it was shown tha t a com pac t m an ifo ld adm itt ing a cohom ologically pos i t ive line bundle is a lg e b ra ic , it m a y be shown tha t a m anifo ld adm itt ing a cohom olog ica lly pos i t ive v e c to r bundle ad m its an em bedd ing in a G ra s sm a n n m anifo ld (see Ref. [6] ). Since a G ra s s m a n n m anifo ld is e a s i ly shown to be a lg e b ra ic , it follows tha t a m anifo ld adm it t ing a cohom olog ica lly p osit ive v e c to r bundle is a l so a lg e b ra ic .

IAEA-SM R-11 /2 3 181

Definition:

E is s a id to be p o s i t iv e , if th e re ex is ts an H e rm it ia n m e t r i c h fo r E such tha t if R £ C ” (TM* ® TM* ® H o m (E ,E )) is the c u r v a tu re t e n s o r field a s s o c ia te d to h th rough i ts connection and R is defined by

R = h R e C“ (TM* ® TM* ® E* ®E*)

Then R is a pos i t ive defin i te fo rm on dec o m p o sa b le t e n s o r s of T ® E.T hat i s , in lo c a l c o - o r d in a t e s , R (C ® r¡ ) = Rq is pos it ive

def in i te in the v a r ia b le s Ç and rj.

As ide

F o r d e ta i l s on H e rm it ia n m e t r i c s , com plex connec t ions , e tc . we r e f e r to Ref. [20] . H ere we sh a l l ju s t r e m a r k one o r two points about the defin ition of H e rm it ia n m e t r i c s and the no ta tion u se d above. An H e rm i t ia n m e t r i c on the bundle E is a sm ooth ly defined s e t of H e rm it ia n in n e r p ro d u c ts on the v e c to r sp a c e s E x , one fo r ea ch x e M. Thus h w ill be a s y m m e t r i c l in e a ri s o m o rp h is m h : E ------»- E* o r a sec t ion of E* ® E* with the a p p ro p r ia tesy m m e tr y and pos i t iv i ty p r o p e r t ie s (f ib rew ise) . H om (E ,E ) denotes the v e c to r bundle whose fib re at x is Ь (Е х ,Ех) = E x ® Ex . Thus H om (E ,E ) s E* ® E. We in troduce R = hR so as to be able to r e g a r d it as a qu ad ra t ic fo rm on T ® E. TM (or ju s t T), TM denote the ho lom orph ic and an t iho lom orph ic tangent bundle of M, re sp e c t iv e ly .

We r e m a r k tha t if E is pos i t ive then ЛГЕ is a lso pos i t ive , w here r = d im E , s ince the induced c u rv a tu re on ЛГЕ is ju s t t r a c e R ([20] ) which is t r iv ia l ly shown to be posit ive . Hence a m anifold adm itt ing a pos i t ive v e c to r bundle ad m its a posit ive line bundle and hence is a lg eb ra ic by s ta n d a rd r e s u l t s (see , for example, Ref. [21] ). We sh a l l show l a t e r that th is r e s u l t a l so follows by prov ing that pos i t ive im p lies weakly posit ive .

Definition:

E —2-*- M is sa id to be am ple if

1. The g loba l s e c t io n s ofE g e n e ra te each fib re of E.2. The n a tu ra l m ap -*- E z ® Tz* is onto. H ere Fz is the su b se t of

se c t io n s of E van ish ing at z.

T h is is the s t r o n g e s t def in i t ion of pos i t iv i ty and e x p r e s s e s the fac t tha t , at l e a s t up to f i r s t o r d e r , th e re a r e p len ty of se c t io n s of E.

F in a l ly , though we w ill not m a k e any app l ica tion h e r e , we give the topo log ica l defin ition of posit iv i ty :

182 FIELD

Definition:

E —3— M is sa id to be n u m e r ic a l ly pos i t ive if, fo r a l l com plex ana ly tic su b s e ts W of M and a l l quotient bundles Q of E | W we have:

/ P (C j, . . . , Cs) > 0w

w h ere P is a p o s i t iv e po lynom ial ,R ef. [12], in the C he rn c l a s s e s of Q.

R e m a rk :

The above defin ition of n u m e r i c a l pos it iv i ty is p o ss ib ly the l e a s t w ell u n d e rs to o d of the p os i t iv i ty defin i t ions . G riff i ths r e m a r k s in Ref. [12] that the def in i t ion of a pos i t ive po lynom ial in p a r t i c u l a r m a y need to be r e v ise d . We give h e r e , how ever, an exam ple to i l l u s t r a te the u se of C hern c l a s s e s in the s im p le s t c a s e when E is a l ine bundle. If E is a line bundle H o m (E ,E ) = Œ, the t r i v i a l bundle , and to say tha t E is pos it ive is to say th a t th e re e x is ts an e le m e n t w e C " (T M * ® TM*), which is c lo sed u n d e r d and is p o s i t iv e defin i te . We then o b se rv e tha t we have an exac t sh e a f sequence

0 ------------------------------------------- * '°— M M

and the following p o r t ion of i ts a s so c ia te d exac t cohomology sequences :

. . ----- » НМ.0*) H2(M, Z ) -------► . .

The f i r s t C h e rn c la s s of the line bundle E is then defined as-6(E) andis u su a l ly denoted by c ^ E ) . Noting th a t the inc lu s ion j : Z ------*• (C induces am a p on cohom ology j :H 2( M , Z ) ------»■ H 2(M,<C) and m ak ing the u su a l iden tif ication, v ia De R h a m 's th e o re m , of H2(M, (D) with the De Rham cohomology g roups of d i f fe re n t ia l f o rm s we m ay then show that a line bundle E is pos it ive if and only if jc j(E) m a y b e r e p r e s e n te d by a r e a l (1 ,1) fo rm w e C " (TM* ® TM*) which is pos i t ive defin i te and s y m m e tr ic . That is, jc ^ E ) is cohomologous to w. The p roo f of th is i s , how ever, n o n - t r iv ia l and r e q u i r e s h a rm o n ic theo ry ,Ref. [21].

In Ref. [12] , G riff iths p ro v es v a r io u s r e la t io n s be tw een the d if fe ren t fo rm s of pos i t iv i ty d e s c r ib e d above. T hese r e la t io n s show, am ongst o the r th ings , that a l l types of pos i t iv i ty co incide prov ided we take a suffic ien tly high s y m m e t r i c t e n s o r pow er of the bundle. In th is se c t io n , we sh a l l d i s cu s s and p rove so m e of the r e la t io n s between the d if fe ren t types of pos it iv ity

IA EA -SM R-11/23 183

a s w ell a s d i s c u s s in g the type of van ish in g th e o re m tha t can a r i s e in th is con tex t to i l l u s t r a t e the techn iques used .

Suppose tha t E is a h o lom orph ic v e c to r bundle on M. A sso c ia te d to E we m a y c o n s t ru c t a f ib re bundle P (E ) on M, the p ro je c t iv e bundle of E. The f ib re of P (E) at x is ju s t P (E X), the p ro je c t iv e sp a ce of E x .

P r o b le m s about p os i t ive v e c to r bund les E m a y u su a l ly be t r a n s f e r r e d to ques t ions about pos i t ive l ine bundles by m a k in g the o b se rv a t io n tha t wehave a ca n o n ic a l l ine bundle H ------*-P (E*) and tha t H r e f l e c t s the p os i t iv i typ r o p e r t i e s of E . H is defined by the r e q u i r e m e n t tha t н | P(E*)x is the h y p e r p lane s e c t io n bundle of the p ro je c t iv e sp a c e P (E*)X. A lte rn a t iv e ly , H is the d ual of the line bundle a s so c ia te d to the p r in c ip a l bundle E * -M ---- »- P(E*).

We then have the following r e s u l t s r e la t in g p os i t iv i ty p r o p e r t i e s of H and E:

1. H is cohom olog ica lly pos i t ive (am ple , weakly posit ive) if and only ifE is cohom olog ica lly pos i t ive (am ple, weakly posit ive) .

2. E p o s i t iv e im p l ie s H posit ive .

The p ro o f tha t E p o s i t iv e im p l ie s H pos i t ive is e a s i ly done by a loca lcom putat ion .

The equ iva lence of the weak p o s i t iv i ty of E and H follows e a s i ly by no ting tha t E * -M з H *-P (E *) and applying the defin ition . We note a l so tha t p o s i t iv i ty im p l ie s weak posit iv i ty . T h is is b e s t s e e n by us ing the m e t r i c on E to define the n o rm function ф = | | 2 : E ------»- R and then noting that a neighbourhood of fixed ra d iu s of the z e ro sec t ion of E is s t ro n g ly p seudo-convex by com puting d i r e c t ly the L evi fo rm L (ф) = 3 5ф. We r e f e r to Ref. [12] for m o r e d e ta i l s on the p roofs of the above im p lica tions .

G riff i ths p ro v e s the following g en e ra l iz a t io n of K o d a i r a s 's van ish ing th e o re m :

T h eo rem :

If E ------»- M is g e n e ra te d by i ts s e c t io n s and if F ------► M is a line bundlesuch tha t F* ® К ® det E is nega tive (K is the can o n ica l bundle Лш T*M) then:

Hq(M, E_(s) ® F) = 0 q > 0 and s г 0

We s h a l l p ro v e h e r e the c l a s s i c a l K o d a ira th e o re m p a r t ly :

T h e o re m :

If E ------»- M is a nega tive l ine bundle then

Н Ч(М, £2P (E)) = 0 fo r p+q< n

C o ro l la ry (K odaira van ish ing the o re m )

Hq(M,_E) = 0 fo r q> 0 if E ®.K* is pos it ive .

184 FIELD

T he p ro o f of the c o r o l l a r y follows by S e r r e duality, Refs [21, 30] :

НЧ(М ,Е) = НП"Ч(М, К ® E*)

P ro o f of th e o re m :

Suppose we have an H e rm it ia n m e t r i c h on E and a K â h le r m e t r i c on M with K â h le r fo rm H (for ex am ple , - i / 2 © £ , w here ©E is the c u r v a tu re te n s o r f ie ld of h on E).

We have the in n e r p roduc t defined on E -v a lu e d (p ,q) fo rm s :

(ф,ф)= ^ С Р,Ч(М ,Е )м

(H ere * is the Hodge s t a r o p e r a to r and #: E ------*- E* is the conjugate l in e a ri s o m o rp h i s m induced f ro m h. )

Let S> = _ The n 2& is the ad jo in t of Э with r e s p e c t to the above in n e r p roduct:

(Ъф,ф) = (ф, &>ф) fo r ф е С р,ч"1(М ,Е ) and ф е ( ? ' q (M ,E)

We le t □ = 3££> + &>d denote the com plex L a p l a c e - B e l t r a m i o p e r a to r ,□ : СР,Ч(М, E)----- ►- Cp,q(M, E).

Set Hp,q(E) = {ф: Пф = 0} ={ф:дф = Ш>ф = 0} С Cp,q(M ,E ). Then НР,Ч(Е) isca l le d the sp a ce of h a rm o n ic (p, q) E -v a lu e d f o rm s on M.

We now define an o p e r a to r of g re a t im p o ra tn c e in the study of K âh le r m an ifo ld s and in H odge 's w ork in a lg e b ra ic geom etry , Ref. [32] :

L et L : Cp,q(M, E ) ---- ► Cp+1,q+1 (M, E ) be defined by L(</>) = H л </>.A lso s e t Л= L*, the adjoin t of L. Then Л: СР,Ч(М, E ) -► Cp' 1'q"1(M, E).We now need the Nakano inequa l i t ie s :If ф e Hp,q(E), then

i /2 (А(©}л ф,ф) SO........................ A

i /2 & л А ( ф ) s 0

H ere © d e n o te s © E, the c u r v a tu re fo rm of h. T h e se in e q u a l i t ie s a re va l id when M is a K â h le r m anifo ld . We sh a l l not p ro v e th e m h e r e (we r e f e r to Ref. [21] ),but m e r e ly r e m a r k tha t they follow s t r a ig h t fo rw a rd ly , without lo c a l co m pu ta t ions , f ro m the following b a s ic eq u a l i t ie s fo r a K â h le r m anifo ld :

Э Л - Л Э = Ш ; З Л -Л Э = Ш ; (ЭЭ + ЭЭ ) ф = @ Л ф

.Suppose then th a t Е is nega tive and we have H=_i/2©.The Nakano in e q u a l i t ie s , A, im ply e a s i ly that:

( (A L - L A )ф,ф) s 0 В

IAEA-S M R-11 /2 3 185

On the o th e r hand,we have the e le m e n ta ry iden ti ty (proof in Ref. [32] ):

(A L - L A )$ = (n-p-q)........................ С

В and С im ply im m e d ia te ly that:

(n— p— q) (ф, ф) S O ................................D

The next s tep u s e s the only r e a l ly deep m a th e m a t i c s in the proof. F ro m a th e o re m of K o d a ira (which u s e s h a rm o n ic th e o ry — p roo f in Ref. [21] ) it follows that

Hp,q(E) = Hq(M, П р (E))....................E

E and D imply the r e s u l t , s ince (ф,ф) is a lways pos i t ive .

R e m a rk s :

F o r f u r th e r d e ta i l s on the above p roo f we r e f e r to Refs [12, 21] . This th e o re m has m any im portan t app l ica t ions . K o d a ira o r ig ina lly used it to p rove tha t Hodge m anifo lds a r e a lg eb ra ic . His p roo f is given in fu ll in Ref. [21]. A nother im p o r ta n t appl ica t ion is in the w ork of H irz e b ru c h and K o d a ira to the study of com plex s t r u c t u r e s on P n, Ref. [17].

G riff i ths in Ref. [12] a sk s the ques tion w hether one can p rove a p r e c i s e K o d a i ra van ish ing th e o re m , fo r v e c to r bund les , of the form :

If E is pos i t ive then is Hq(M ,E*) = 0 for q s n - r ?He p ro v e s a sp e c ia l c a se of th is co n jec tu re when r=d im E = 2 and E has a nons in g u la r section . Le P o t i e r has s ince p roved th is co n je c tu re in full generality . We r e f e r to a s e m in a r by h im in th e se P ro c e e d in g s .

A nother r e s u l t in th is d i r e c t io n in Ref. [12] is the following:Suppose E ------»-M is p o s i t iv e and f is a n o n - s in g u la r sec t io n of E with

z e r o locus S. If we blow up M along S to obta in M then M-S m a y be p ro v ed to be n - r+ 1 p se u d o -c o n v ex (see s e c t io n 3). U sing th is fac t G riff i th s p ro v e s tha t if I deno tes the id e a l sh e a f of S and F is a ho lom orph ic v e c to r bundle on M sa t is fy in g a su i tab le p o s i t iv i ty condition then th e re e x i s ts k(F) such that:

Hq( M ,I k® F ) = 0 for k ê k ( F ) and q S n - r

L e t u s now show how, u s in g the p r o je c t iv e bundle P (E *) , E posit ive im p l ie s tha t E is cohom olog ica lly pos i t ive . The p roof w ill a l so give the equ iva lence of cohom olog ica l p o s i t iv i ty of E and H.

Suppose then tha t H ------- P(E*) = P is the can o n ica l l ine bundle d e s c r ib e dabove. L et ^ be a c o h e re n t sh e a f on M. We a s s e r t that H q( M , I ï ^ ® &)~ BPiPj^H15 ® 7г*,§г ); the r e s u l t then follows im m e d ia te ly f ro m the r e s u l t fo r l ine bund les .

1 8 6 FIELD

R e c a ll that the p-th derived sh eaf Rp (Hk ® ir* & ) of Hk 0 тг*, is the sh eaf on M a sso c ia te d to the p re sh ea f:

U -----► HP(7Г-1 (XJ), H*1 ® v*3r I тг' 1 (U))

We now m ake u se of the fact that n is p ro p er to give the re su lt that the n atu ra l m ap:

R ^ H 1' ® г * З П ( х ) -------► H P (H k ® I tt'^ x ))

is an iso m o rph ism (see Ref. [9] ).That is :

RP ( l í S^\ir'\x))

w here ^ is the tr iv ia l sh e a f over 7г"1(х) = P (E *) з P r_1, and H is the hyperplane sectio n bundle of P (E *) .

But н р(р г" \ н к) = 0 p> 0

~E¿k) fo r p = 0

(this is why we u se the sy m m etric product) and it follow s that

RP(H ® 7rS|i S'” )x = 0 ,p> 0

« (E (k) ® 5F)x,p = 0

It now follow s without d ifficu lty that

R°(H k ® тг* S') « E (k) ® S7"

An application of the L e ra y sp e c tra l seq u en ce ,R ef. [9], g iv e s the re su lt im m ediately .

F in a lly , a s another exam ple of the function-theoretic app lication s of van ish in g cohom ology, we prove that if E is cohom ologically p o sitiv e then there e x is t s k> 0 such that E ® is am ple.

P ro o f

L et zQS M and be the id ea l sh eaf of z 0. Then there e x is tsk (z0) 6 Z + such that

Hq(M, J ® E (k(Zo,) ) = НЧ( У ® E (k(Z<|))) = 0 fo r q> 0 z0 — z0 —

IAEA-SMR-11/23 187

F ro m the cohom ology exact seq u en ces of

0 ® E (k)-z 0 — E

0 -У®Е Z0 -- — z . z.

0

we obtain

Z

fo r k ê k ( z 0). The coherence of the sh eav es im p lie s that there e x is t s a neighbourhood U (z0) of z0 such that Z holds in U(z0) fo r k = k(z0). We then o b serv e if Z holds fo r z e U ( z 0) then Z holds fo r a ll positive in te g ra l m u ltip les of k (z0). We m ay then rep eat the above p r o c e s s and, u sin g an e lem en tary com pactn ess argum ent, find к such that E ^ is am ple.

In a p ap er of th is length we have, of n e c e ss ity , been fo rced to om it m any to p ics c lo se ly re la ted to the above m a te r ia l. In p a r t ic u la r , we have not r e fe r re d to the theory of d eform ation s of com plex s tru c tu re s , R efs [2 1 ,2 2 ] . G riffith s ' p ap er on the extension problem in com plex a n a ly s is ,R e f . [13], a lso p ro v id es a u se fu l so u rce of exam p les and p ro b lem s of a sh ea f coh om ological type — at le a s t in the fo rm al theory.

[1] ANDREOTTI, A ., T héo rèm es de dépendance a lg éb riq u e sur les espaces com plexes pseudo-concaves,B u ll.S o c . M ath .F ran ce 91 (1963) 1.

[2] ANDREOTTI, A ., GRAUERT, H ., T héo rèm es de fin itude pour la cohom olog ie des espaces com plexes . Bull. S o c .M a th .F ra n c e 90_(1962) 193.

[3] ANDREOTTI, A ., VESENTINI, E ., C arlem an es tim ates for the L a p lace -B e ltram i eq ua tion on com plex m anifo lds, I .H .E .S . 2 £ 81.

[4] ATIYAH, M ., SEGAL, G ., SINGER, Papers on the A tiyah -S inger Index T h eo rem in A n n .M a th ., in p a rticu la r I, II, III, 87 (1968) 531.

[5] BREMERMANN, H .J . , Úber d ie Â qu ivalenz der p seudo-konvexen G ebie te und der H o lom orph iegeb ie te im Raum von n kom plexen V erà'nderlichen, M ath .A n n . 128 (1954) 63.

[6] CHILLINGWORTH, D ., Sm ooth m anifo lds and m aps (these P roceed ings).[7] DOUADY, A ., e t a l . , T op ics in sev era l co m p lex variab les, M onographie N o. 17 de L' E nseignem ent

M athém atique , G enève (1968).[8] FIELD, M .J . , H o lom orph ic function theory and co m plex m anifo lds (these P roceedings).[9] GODEMENT, R ., T opo log ie A lgébrique e t T h éo rie des F aisceaux , H erm ann, Paris (1964).

[10] GRAUERT, H ., On L e v i's P roblem and th e em bedd ing of r e a l a n a ly tic m an ifo lds, A n n .M a th . 68 (1958) 4 60 .

[11] GRAUERT, H ., lib e r M odifika tionen und E x z ep tionelle A naly tische M engen, M ath .A n n . 146 (1962) 331.[12] GRIFFITHS, P .A . , H erm itian d iffe re n tia l geo m etry , C hern classes and positive vec to r bund le s. In

'G lo b a l A n a ly s is ', P rinceton U niversity Press (1969).[13] GRIFFITHS, P .A . , T h e ex tension p rob lem in com plex analysis II, A m .J . M ath . ji8 (1966) 366.

R E F E R E N C E S

188 FIELD

[14] GUNNING, C .R ., ROSSI, H ., A n a ly tic Functions of S evera l C om plex V ariables, P ren tice -H all,Englewood C liffs, N .J . (1965).

[15] HARTSHORNE, R ., "A m ple v ec to r bund les", I . H .E .S . 29 (1966) 319.[16] HIRZEBRUCH, F . , T o p o lo g ica l M ethods in A lgeb ra ic G eom etry , S pringer (1966); (w ith appendixes

by R .L .E . SCHWARZENBERGER and A . BOREL).[17] HIRZEBRUCH, F . , KODAIRA, K ., On the co m plex p ro je c tiv e spaces, J. M ath . Pures A ppl. 36 (1957) 201.[18] HÔRMANDER, L ., An In troduc tion to C om plex A nalysis in S evera l V ariables, Van N ostrand (1966).[19] HÔRMANDER, L -, L inear P artia l D iffe ren tia l O perators, S pringer (1964).[20] KOBAYASHI, S . , NOM IZU, K ., F oundations o f D iffe ren tia l G eom etry , 1_ and 2 , W iley In te rsc ience ,

New York (1963).[21] KODAIRA, K ., MORROW, J . , C om plex m anifo lds, H olt, R inehart and W inston I n c . , New York (1971).[22] KODAIRA, K ., SPENCER D .C . , On deform ations of com p lex a n a ly tic structures I and II, A nn. M ath . 67

(1958) 328 ; and also III, A nn. M ath. 7 ^ (1960 ) 43 .[23] KOHN, J . J . , H arm onic in teg ra ls on strongly pseudoconvex m anifo lds I and II, A n n .M a th . 78 (1963) 112.[24] KOHN, J . J . , "Boundaries o f co m p lex m an ifo ld s’’, P ro c .C o n f.C o m p lex A nalysis, M inneapolis, Springer-

V erlag (1965).[25] LEWIS, М ., An in e ffec tiv e w eakly n o n -lin e a r u n ren o rm aliz ab le po ly n o m ia l Lagrangian , I . C .T .P . ,

T rie s te , p rep rin t (1970).[26] NORGUET, F . , Sur les dom aines d 'h o lo m o rp h ie des fonctions uniform es de plusieurs variab les com plexes

(passage du lo c a l au g loba l), B u ll.S oc . M ath. F rance 82 (1954) 137.[27] ОКА, K ., "D om aines pseudoconvexes” , T ôkiku M a th .J . 49 (1942) 15 . a lso "Lem m e fo n d am en ta l" , J.

M a th .S o c .Japan 3_(1951) 204 and 259.[28] ROSSI, H ., S trongly pseudoconvex m anifo lds, Springer Lecture notes N o .140 (1972).[29] S em in a ire CARTAN, H ., 4 (1951-1952), B enjam in , New York (1967); 5 (1953-1954), B enjam in,

New York (1967).[30] SERRE, J .P . , Un th é o rèm e de d u a lité , C o m m e n t.M a th .H e lv . 2 £ (1955 ) 9.[31] VESENTINI, E ., On Levi convex iv ity and cohom ology vanish ing theorem s, T a ta In stitu te , Bombay (1967).[32] WEIL, A ., V arie ties K ah leriennes, H erm ann, Paris (1958).

IAEA-SMR-11/24

COMPLEX ANALYSIS ON BANACH SPACES

M .J. FIELD M athem atics Institute,U niversity o f Warwick,Coventry, Warks,United Kingdom

Abstract

COMPLEX ANALYSIS ON BANACH SPACES.A b rie f survey o f re c e n t results in com p lex analysis on Banach spaces is g iven .

In this p ap er, we w ish to give a b r ie f su rv ey o f som e recen t developm ents in the theory of com plex a n a ly s is on Banach sp a c e s . We sh a ll m ake no attem pt to m ake a com plete su rv ey o r provide an ex ten sive bibliography a s a su rv ey a r t ic le by Leopoldo N achbin should sh ortly be appearin g in the B u lletin o f the A m erican M ath em atical Society .

R e c a ll that a com plex-valued function f defined on an open su b se t U o f the com plex B an ach sp ace E is sa id to be analytic i f f i s once continuously d iffe ren tiab le on U with d eriv ative a com plex lin e a r m ap:

Df: U - L c (E, Œ)

It is w ell known [1 , 12] that th is definition i s equivalent to any o f the following sta tem en ts:

(a) f adm its a convergent pow er s e r i e s rep re sen ta tio n at ev ery point o f U .(b) f i s continuous and f r e s t r ic te d to com plex lin es i s analytic a s a m ap

from Œ to Ф .(c) A s fo r (b) except a ssu m e that f is continuous at only one point in each

connected com ponent o f U .(d) A s fo r (b) but a ssu m e that f i s bounded at only one point of each

connected com ponent o f U .

R em ark s

1. If in p rop erty (b) above we com pletely drop the continuity condition on f we sa y f is G âteaux an aly tic . With the continuity condition f is so m e tim e s ca lled Frfechet analytic .

2. U sing, e .g . p rop erty (a) the notion o f analytic functions can be defined and stud ied fo r m o re g e n e ra l top o log ical v ec to r sp a c e s , in p a r t icu la r , lo ca lly convex top o log ica l v ec to r sp a c e s . We r e fe r the in terested r e a d e r to R ef. [ 1] fo r a developm ent fo r the n o n -B an ach -space c a se .

The study o f analytic functions defined on in fin ite-d im en sion al v ecto r sp a c e s i s old, having its roo ts in w ork o f F rë c h e t and G âteaux in the 1900 's. In te re st in the theory in the p a st decade has been stim u lated fo r se v e ra l r e a so n s . In the f i r s t p lace , B a n ach -sp ace com plex an a ly s is techniques have proved valuab le in the study of e sse n tia lly fin ite-d im en sio n al p ro b lem s.

189

190 FIELD

In p a r t ic u la r , we m ention the work of Douady on the problem o f m oduli [ 9](this p ap er p ro v id es the stan dard re fe ren ce fo r the b a s ic theory and d efinitions on com plex B anach m an ifo lds). Secondly, w ork o f E e l l s and Elw orthy, following on w ork o f K u iper, B u rgh lea , M oulis and o th ers on in fin ited im en sion al d iffe ren tia l topology, show s that every se p a ra b le H ilbert m anifold adm its a com plex stru c tu re (in fact, many'. See R ef. [3 ] ) .E x am p le s o f such m an ifo lds include Sobolev c la s s e s o f m ap s from a com pact d ifferen tiab le m anifold to another d ifferen tiab le m anifold. K rik orian (T h esis at C orn ell) using id e a s o f Douady has shown that Cr (M ,N ), r ë 0, adm its a n atu ral com plex stru c tu re when N is a com plex m anifold. The g e n era l in te re st shown in in fin ite-d im en sion al d iffe ren tia l topology has a lso provided a stim u lu s to develop a correspon din g theory fo r the analytic c a se .One featu re h ere i s that m any B anach sp a c e s , in p a r t ic u la r C[ 0, 1] (continuous functions on the unit in terval) do not adm it any d ifferen tiab le bump functions, though they do have, o f co u rse , plenty o f analytic functions (with non-bounded support) on them . Thus, in som e c a s e s it m ay be m ore natural to w ork with an alytic functions and techniques than d ifferen tiab le on es.

H aving d isc u sse d a little of the m otivation and h istory o f the su b ject we now turn to a d isc u ss io n o f som e o f the p ro b lem s and the re su lts so fa r obtained.

Noting that the p rin c ip le o f an alytic continuation s t i l l holds in Banach sp a c e s a v e ry n atu ra l p rob lem is to g e n e ra lize som e of the r e su lt s about holom orphy dom ains that hold when E i s fin ite-d im en sio n al. However, new p ro b lem s a re faced when one t r ie s to form u late a good definition o f "dom ain o f holom orphy" . F o r exam ple , in fin ite d im ensions a dom ain of holom orphy is alw ays the dom ain of ex isten ce o f an analytic function (see R ef. [ 10 ], fo r p roof). T h is is no lon ger so in infinite d im ensions [ 13] .One o f the m ain con struction al too ls used in finite d im ensions — fillin g up the dom ain with a countable se t o f c lo sed and bounded su b se ts and constructing an alytic functions inductively — no lon ger w orks fo r the re a so n that analytic functions w ill not, in gen era l, be bounded on a c lo sed and bounded se t u n less it i s com pact and one cannot f i ll up open subdom ains o f an in fin ite-d im en sion al B anach sp ace by a countable se t o f com pact su b se ts (E is not lo ca lly com pact). T h is le a d s to the study o f sp e c ia l c la s s e s of analytic functions on a given subdom ain £2 o f E , fo r exam ple , the se t o f analytic functions which are bounded on a ll c lo sed and bounded su b se ts o f Г2. F o r th is c la s s o f functions, r e su lt s o f the C artan -T h iillen type have been proved [ 7 ].

A problem , sp e c ia l to the in fin ite-d im en sion al c a se , o ccu rs when we a sk how fa r a dom ain o f holom orphy can be d e sc r ib ed in te rm s o f its in te rsec tio n s with fin ite-d im en sio n al (linear) s p a c e s . F o r exam ple , i f i i c E p o s s e s s e s p rop erty X , w here X m ight be: p seudo-con vex, dom ain of ex isten ce , dom ain of holom orphy, e t c . , then fi n (Dm a lso p o s s e s s e s p ro p erty X . The re v e r se im p lication i s m uch m ore d ifficu lt, but r e su lts have been obtained when fi s a t i s f ie s som e polynom ial convexity conditions [19, 21].

G iven an a rb itra ry open su b se t f i c E one can co n stru ct its envelope o f holom orphy and p rove uniqueness in a m anner s im ila r to that done in R ef. [ 10] . We r e fe r to R ef. [ 14] fo r a full treatm en t. M ore d ifficu lt i s the re la tion between the envelope o f holom orphy constructed v ia sh eav es and the sp ectru m of A(fi). The sp ec tru m o f A(fi) w as f i r s t studied by C oeuré in R ef. [ 5 ] .I understand that equ ivalen ce between the two approach es h as now been attained, at le a s t in som e c a s e s .

lAEA-SMR-11/24 191

In C o e u ré 's study o f A(Q) he is faced with the p rob lem of topologizing A(U) a s a topo logical v ec to r sp a c e . The com pact open topology ap p ears not to be a su itab le topology fo r studying p ro b lem s of analytic extension and he co n stru c ts a topology su itab le fo r study o f an aly tic extension . Study o f the v ec to r sp ace A(S7) le ad s to in triguing p ro b lem s in functional an a ly s is a s so c ia te d with the fac t that there a re s e v e r a l n atu ral topo logies that one can put on A(Q). One featu re o f th ese topo log ies i s that they gen era lly have the sam e bornology. We r e fe r to [ 6 , 17] fo r fu rth er d e ta ils .

Another in terestin g a r e a o f developm ent has been in w ork o f Gupta [11] and Nachbin on sp a c e s o f n u clear an aly tic m a p s . R e su lts have been obtained h ere gen era liz in g w ork of M artin eau [ 16] and M algran ge [15] on convolution o p e ra to r s . A n u c lear analytic m ap is e sse n tia lly a n o n -lin ear gen era lizatio n o f a t r a c e - c la s s o p era to r and is stro n g ly approx im atab le by an aly tic m ap s o f fin ite ran k . Other c la s s e s o f an alytic m ap s have a lso been studied , in p a r t ic u la r by Boland and D w yker. H irsch o w itz 's con struction fo r the envelope o f holom orphy allow s one to con struct a n u clear envelope o f holom orphy, m ax im al fo r an alytic continuations o f n u c lear analytic m ap s.

P ro b ab ly the m o st productive a re a in in fin ite-d im en sion al com plex a n a ly s is has been in the a lg e b ra ic study o f the lo c a l r in g s 0 O(E) and an aly tic s e t s .

To obtain a sa t is fa c to r y theory, one has to a ssu m e that o n e 's analytic s e t s a re defined by fin itely m any analytic functions lo ca lly (exam ples by Douady and oth ers show the im portan ce o f th is re str ic tio n : every com pact m etr iz ab le top o log ica l sp a c e m ay be rep re se n te d a s an analytic se t if the re s tr ic t io n on fin iten ess is rem oved).

The prob lem is then to study the re la tio n betw een an alytic su b se ts o f E and id e a ls o f 0O(E). W ork in th is a r e a has been done by R am is , Ruget,M azet and o th ers and is w ell convered in R a m is ' m onograph [ 20 ].

C h ief am ong the r e su lt s a re ch arac te r iz a tio n s o f the id e a ls that c o r r e spond to an aly tic s e t s defined by fin itely m any an aly tic functions, the n u llste llen sa tz fo r such an aly tic s e t s and gen era liza tio n s o f R em m ert- S te in 's th eorem and Chow 's theorem to in finitely m any v a r ia b le s . P ro b lem s outstanding involve finding coherence p ro p e rtie s o f 0O(E) and a deeper study of the lo c a l ring 0 O(E) with a view to g en era liz in g O k a 's n orm alization theorem .

R E F E R E N C E S

[ 1] BOCHNAK, SICIAK, papers in S tudia M a th e m a tic a on a n a ly tic functions on to p o lo g ica l vec to r spaces (1970 -) .

[ 2 ] BREMERMANN, H. J . , H olom orphic functionals and com plex convex iv ity in Banach spaces, P ac . J. M ath . 7 (1957) 811.

[3 ] BURGHELEA, D ., DUMA, A . , A n a ly tic com p lex structures on H ilbert m an ifo ld s, J. D iff. G eom .5 (1971) 371.

[4 ] CARTAN, H . , S em in a ire Bourbaki.[ 5 ] COEURE, G ., T hesis, A nn. Inst. F ourier, 20 1 (1970) 361.[ 6] DINEEN, S . , H olom orphy types on a Banach sp ace , S tudia M ath . (1971).[ 7 ] DINEEN, S . , "T h e C a r ta n -T h u lle n theo rem for Banach spaces” , A nn. Sc. N orm . Super. Pisa 24

4 (1 9 7 0 ) 667 (co rrec tio n fo llow ing).[ 8 ] DINEEN, S . , HIRSCHOW ITZ, A . , "Sur le theo rem de L ev i-B anach", C . R. 272 (1971) 245.[ 9 ] DOUADY, A . , "Les p rob lèm es des m odules . . Ann. Inst. Fourier 16 1 (1966) 1.

[1 0 ] FIELD, M . J . , H o lom orphic function theory and co m plex m anifo lds, these P roceedings.

[1 1 ] GUPTA, С . , M alg range theo rem for n u c lea rly en tire functions o f bounded type on a Banach space ,Notas de M a th e m a tic a , N o. 37 , Institu to de M a th e m a tic a Pura e a p lic a d a , Rio de Janeiro .

[1 2 ] HILLE, E .# PHILLIPS, R .S . , F unctional analysis and sem i-g roups, A m er. M a th . Soc. C o lloq . Publ.31 (1957).

[ 13] HIRSCHOWITZ, A . , "R em arque sur les ouverts d 'h o lo m o rp h ie d 'un p rodu it denom brab le de d ro ites" , A nn. Inst. F ourier, 19 (1969) 219.

[1 4 ] HIRSCHOWITZ, A . , "P ro longem en t an a ly tiq u e en d im ension in f in ie " , C .R . 270 (1970) 1736.[1 5 ] MALGRANGE, B ., "E x istence e t ap p rox im ation des solutions des équa tions aux dérivées p a rtie lle s et

des équa tions des convolu tions", Ann. Inst. Fourier £ (1 9 5 5 ) 271.[1 6 ] MARTINEAU, A . , "Sur les fonctionnelles ana ly tiques e t la tran sfo rm ation d e F ourier-B orel" , J. d 'A n a l.

M ath . 11 (1963) 1.[1 7 ] NACHBIN, L . , T opology on Spaces o f H olom orphic M appings, S pringer-V erlag (1969).[1 8 ] NOVERRAZ, P h . , T hesis, A nn. Inst. Fourier 19^2 (1969) 419.[1 9 ] NOVERRAZ, P h . , "Sur la co n v ex ité fo n c tio n n e lle dans les espaces de B anach", C .R . 272 (1971) 1564.[2 0 ] RAMIS, J . P . , Sous-ensem bles ana ly tiques d ’une v a r ié té an a ly tiq u e co m p lex e , S p ringer-V erlag (1970).[2 1 ] NOVERRAZ, P h ., P seudo -C onvex ite , C onvex ite P o lynom iale e t D om aines d 'H o lom orph ie en D im ension

In f in ie , N orth -H o lland (1973).

192 FIELD

IAEA-SMR-11/25

MODERN THEORY OF BILLIARDS - AN INTRODUCTION

G. GALLAVOTTI Istituto M atem atico,U niversité di Roma,Rome, Italy

Abstract

MODERN THEORY ОГ BILLIARDS - AN INTRODUCTION.This paper presents an e lem en ta ry in troduction to the notions and ideas involved in th e proof o f the

e rg o d ic ity o f b illia rd s .

1. B ILLIA R D S, DO YOU RECO GN IZE IT ?

L et N = [ 0, 1] X [0 ,1 ] mod 1 be a tw o-d im ensional to ru s . L e t Q j, Q2,. . . , Qm be m c lo sed convex reg io n s in N. A ssu m e that Qj i s a C2-sm oothcurve with n ever van ish ing curvature and a ssu m e a lso that Qj П Q = ф, i f Í-

Let V be the R iem annian m anifold (with boundary) obtained by taking out of N the in te r io r o f Q j, . . . , Qm; the m etric on V i s the one inherited by N ( i . e . d s 2 = dx2 + dy2).

L e t T jV be the unitary tangent bundle of V.The e lem en ts of T jV can be thought of a s applied v e c to rs o r a s couples

(q, 6), qeV and 0 s 0 S 2 n, where qGV is the point of app lication of thev ec to r and в is its angle with a fixed d irection .

Define on TjV a probability m e a su re p (d q d 0) = (const) ■ dq dS and a flow St , — o c< t< + oc, S t : T jV - T ¿V. T h is flow is defined a lm o st everyw here and is con structed by m ean s of the b ill ia rd s ru le a s i s shown in F ig . 1 (where the c a se t >0 is con sid ered):

One c le a r ly re co g n ize s in the dynam ical sy stem (T jV , St , ¡j.) an "o rd in a ry " gam e of b i l l ia rd s with one b a ll on a perio d ic tab le with m o b s ta c le s . The m e a su re p i s p re se rv e d by S j .

Now, the follow ing theorem holds:T h eorem (Sinai): The dynam ical sy ste m (T: V, St , ц) i s ergod ic and,

m ore p re c ise ly , a K - sy ste m .To attack the problem of the proof of the th eorem , f i r s t rem ark that the

flow S, can be m ore sim ply d e scrib ed through a " se c tio n " of i t s e lf . To d isc u s s th is point and the follow ing on es, let us choose, from now on, a g e o m etr ica lly sim p le gam e of b i l l ia r d s , i . e . let u s co n sid er the ca se in which th ere-is ju st one c ir c u la r Q (with ra d iu s R ).

L e t M be the m anifold of the applied v e c to r s with point of application on 3 Q and pointing tow ard s the in te r io r of Q. An elem ent xGQ can be d e sc r ib e d by two co -o rd in a te s x = (r , <p) w here 0 s r < 2 irR i s the clockw ise a b s c i s s a , over 3Q , of the point of app lication of x and <p i s the angle which the v ec to r x subtends with the outer n orm al to 3Q in r : 7r/2 § <p s 3 7r/ 2.The ran ge of v a lu e s of <p r e f le c ts the fact that M c o n sis ts of v e c to r s "heading" ag a in st 3 Q.

193

GALLAVOTTI

qq' + qq = *

x' = (q; 9 ) =s X = ( q . 6 )

FIG. 1. Construction of flow.

: = (г,фГ

= (г,'Ф'1

t‘ = Tx

( Û o\VV П$) ^ ^ о00FIG.2. Transformation T.

IAEA-SMR-11/25 195

L e t us define a tran sfo rm atio n T : M -*• M a s fo llow s: choose xeM , think o f it a s the v ecto r d e scrib in g a b illia rd ba ll hitting 3Q and follow th is b a ll back in tim e in the past until, at tim e т (x) < 0, it h its again Э Q in a point x ' = T x (F ig . 2).

It is quite e a sy to check that the m e a su re v (dr d<p) = - (const) co s <p drd i s p re se rv e d by T . The m apping T i s only a lm o st everyw here defined.

T h e re fo re , (M, T , v) i s a dynam ical sy ste m and it can be e a sily im agined that the p ro p e rtie s o f (M, T , v) could be tra n sla te d into p ro p ertie s of (T jV , S t , ц): notice that v i s the n atu ral pro jection of ц on M. F o r th is re a so n , we sh a ll concentrate our attention on (M, T , v) without in sistin g on how to tra n sla te the in form ation on it into in form ation on (T jV , S t , ц).

The sy ste m (M, T , v) i s s t i l l non-sm ooth sin ce it h as a boundary S = (x /x e M , ç (x ) = Я-/2, (3 /2) 7г) (there i s no boundary in r s in ce th is coord in ate i s p eriod ic) and sin gu la rity points which co n sist in the se t T_1S .In F ig . 3, x i s a s in g u la r point for T . One e a s i ly finds that the sin gu larity points lie on sm ooth (C ^ -c u rv e s divided into 8 fa m ilie s (if the rad iu s R of the o b stac le i s not too sm a ll com pared with the sid e o f the to ru s). See F ig . 4 where som e elem en ts of two such fa m ilie s a re drawn.

C le a r ly , the cu rv es in th is figu re a re a lso discontinuity cu rv es for r (x), which, how ever, i s continuous on them from one sid e (denoted + in the figu re ).

A reg ion В which o v e r la p s with the sin gu la rity lin es is blown into p ieces by T (roughly a s m any p iece s a s the num ber of lin es in te rse c te d by B ).

S in a i 's idea i s to prove that, in sp ite of its h orrify in g non -sm ooth n ess, (M, T , v) beh aves much in the sam e way a s a C -sy ste m (Anosov d iffeom orph ism ).

2. A R E L A T E D PR O B LEM

In th is sec tio n , we sh a ll f i r s t d isc u s s the idea behind the proo f of the ergod icity o f С - sy s te m s in a v ery sim p le c a se .

T h is proof w ill be u sed to il lu s tra te the n e c e ssity and the m eaning of the v a r io u s m ath em atical o b jec ts that have to be introduce^! to cope with the p rob lem of b i ll ia rd s (as well a s with the theory of the C -sy ste m s) .

1 9 6 GALLAVOTTI

The com pariso n sy ste m i s going to be the much publicized — in many other p ap e rs o f the P ro ce e d in g s — authom orphism of the to ru s N =[ 0, 1 ] X [ 0, 1 ] mod 1 defined by т (x, y) = (x+y, x+2y) mod 1; se e F ig . 5 . L et e+, e_ be the two e igen v ecto rs o f the m atr ix (J \) anc let -+> = X+1 < 1be the two e igen v alu es. It i s e a sy to check that the d irectio n s e+, e . a re ir ra t io n a l, i . e . the tangent of the angle o f these two d irectio n s with the x - a x is a re ir ra t io n a l n u m b ers. The f i r s t c la im is that the sy ste m (N, T , v dxdy) i s a C -sy s te m and that the contracting and expanding fo lia tio n s f c and f e co n sist of p a ra lle l s tra ig h t lin e s (w rapped on the to ru s and p a ra lle l to e+ and e_). Let us note that the ir ra tio n a lity of the d irectio n s e + and e . im p lie s that each le a f o f the fo liation i s dense in N. In fac t, le t (x, y )e N be a point and let Cçc (x, y) be the stra ig h t line p a ra lle l to e . and p a ss in g through (x, y). L e t (x1, y ' ) £ C g (x, y); it i s c le a r that r n (x, y) and t " ( x \ y 1) w ill be at a d istan ce not exceed ing th e ir d istan ce counted along the line t 11 C ê (x, y) which i s

d n (тп (х ,у ) , т " ( х \ у ' ) ) = X+’ nd ((x ,y ), ( x ',y ') )t " c e (x , y) + c E ( x , y )

~c ~c

S im ila r ly , one p ro v es that the fo liation Çe i s con tractin g in the p ast ( i . e . under t ' 1 ) and that the line elem ents of f e lo ca lly expand in the p ast while the line e lem en ts o f §e lo ca lly expand in the fu ture. The expansion

IAEA-SMR-11/2 5 197

and contraction coeffic ien ts a r e alw ays X+ or X_. T h is fac t , of c o u rse , i s re sp o n sib le fo r the p o ss ib ility of an e a sy treatm en t and v isu a litio n of the above sy ste m .

We now show that (N, t , v ) i s ergod ic using a p roof which, though certa in ly not the s im p le st , is ex trem ely in stru ctiv e sin ce it contains the m ain id ea , due to Hopf, at the b a se of the proof of the ergod icity of C - sy s te m s and b ill ia r d s .

3 . PRO O F OF THE ERG O DICITY OF ( j *).

L e t f be a continuous function fe C (N ) . Then, by the B irkh off theorem , the lim its

e x is t a lm o st everyw here and a re a lm o st everyw here equal: f +(x) = f"(x ) fo r x e U , v(U) = 1.

We have to prove that for a ll feC (N ) the functions f +(x) and f '( x ) a re constant a lm o st every w h ere. T h is, of co u rse , im p lie s the ergod icity of (N, t , v).

C on sid er a covering of N with sq u a re s Ulf U2, . . . . with s id e s not exceed ing 1 Д/2 and p a ra lle l to e+ and e_. We sh all choose the sq u a re s in such a way that they o v erlap in chain ( i . e . if P , Q eN th ere i s a chain U¡ ,

The fam ily {U ¡} can be chosen to contain a finite num ber of e lem en ts. C le ar ly , it w ill be su ffic ien t to prove that f + and f" a re constant on each

Uj (alm ost everyw here).So let u s fix feC (N ) and show that it s a v e ra g e s in the future and in the

p a st a r e a lm o st everyw here constant on U1; sa y .If P E U ! let C Ee(P) be the expanding le a f through P and let (P) be the

connected part of C^ (P)DUi containing P; s im ila r ly , we define ü (P), se e F ig . 6 . e c

In Ui, we u se a sy ste m of orthogonal co -o rd in a te s b a se d on the v e c to rs e+ and e_, which a re p a ra l le l to the s id e s o f U j. If В i s a m easu rab le su bset o f C f (P) or C { (P), we denote by | В | it s L eb esgu e m e a su re with re sp e c t to the a b s c i s s a o r the line; hence, in p a r t ic u la r , | C £ (P) | = | C £ (P) | = length of the sid e o f U j. с с

L e t us now con sid er the set

n

n

i= 0

U•: , . . . , Ui such that v (Ui DU )> 0 and Р е Ц , QeU,- ). i2 Li

’e

w here U = (x /x e N , f+ (x) and f ' (x) e x is t , f+ (x) = f"(x ).

198 GALLAVOTTI

FIG.7, P and Q with local contracting and expanding leaves.

S in ce, by the B irkh off th eorem , v (U) = 1 (see above) it follow s by F u b in i's th eorem that Ve h as fu ll m e asu re i. e . y(Ve ) = y ( U j) .

P ro o f: v (Uj) = v (U nU j), hence i f s and s ' a re the co -o rd in a te s of P e U j,P = (s , s ') , we find

W P)=/ ds,|V ° ’s,)nunui|which im p lie s that fo r a lm o st a ll s ' , we have |C f (0, s 'JO U flU il = |С^ (0, s ' ) and, aga in by F u b in i's th eorem , th is im p lie s that ] C je(x)nUDUi) = 6|c 'ge(x )| fo r a lm o st a l l х е и ц in w ord s, we can say that a lm o st a ll points xE U j a re such that the line Ce (x) l ie s a lm o st en tire ly in U nU j.

S im ila r ly , we can define Vc and show that v (V ) = v (U ^ .L et us now con sid er the set

v (Ux) = id s ' Id s

v = u n u 1nv en v c

C learly , V h as full m e a su re in U jî v (V) = i/ÍU j). It i s , th ere fo re , enough for our p u rp o se s to show that f +(x) = f" (x) = const for x eV .

L e t P , Q eV , then draw through P the lo ca l contracting le a f C f (P) and through Q the lo ca l expanding le a f C Êg(Q) (see F ig . 7). °

T he point T m ay o r m ay not lie in V. In any c a se , it i s p o ssib le tofind a point P 'e C Êc(P) and a point Q 'e C Êe(Q) such that the points T ', P 1,Q1 a re a ll in V (see F ig . 7). In fac t , by con struction , a lm o st a l l points on the two lin e s C êc(P) and C« (Q) lie in V (rem em b er the choice of P and Q), hence a s P ' ru n s o v er V n C {c (P) and Q1 over V n C Se(Q )th e point T ' sp an s a se t o f full m e asu re which, th ere fo re , c e rta in ly in te r se c ts V.

Now, the gam e i s over; in fac t, by construction :

d (rnP , T n P ' ) -------------- 0n-> <*•

d (т~п P ' , т 'пТ ') ------ ► 0П-» Ж-

d ( t 11 T 1, T n Q ’) -----------► 0

d(T’ nQ ', t '"Q ) ----- » 0n-> «.

IAEA-SMR-11/25 199

(with exponential sp eed ). H ence, by the uniform continuity of f,

f ( r n P) - f (rnP ') ------» 0n -*■

f(T ‘ nP ') - f ( r ' nQ ') ----- »• 0 e tc .n-> »

T h e re fo re , f +(P) = f +( P '); f ' ( P ') = f ( T ' ) ; f+ (T ') = f +(Q'); f '(Q ') = f "(Q); but, by construction , it i s a lso tru e that P , P 1, Q, Q1, T 1 a re in VCU hence f +(P) = f '( P ) ; f +(P ') = f ' ( P ') ; f+ (T ') = f '( T ') ; f +(Q') = f '(Q ') ; f+(Q) = f '(Q ) . H ence, a l l the above m entioned v a lu e s o f f + and f" coincide; in p a r ticu la r ,

f +(P) = f+ (Q) = f" (P ) = f ‘ (Q)

which m ean s, by the a rb itr a r ity o f P and Q, that f + and f a re constant onV (and, th ere fo re , a lm o st everyw here).

4 . HOW TO G E N ER A LIZ E THE ABOVE ARGUM ENTS

O bviously , in m ore gen era l s itu a tio n s, things a re not so n ice and e asy ; n e v e rth e le ss , the proof of ergod icity fo r the c a se of С - sy s te m s or even b i l l ia r d s sy s te m s p ro ceed s along the sam e lin es a s the above p roof o f the e rgo d ic ity o f (J gi

lt i s p o ss ib le , in those c a s e s , to con struct a denum erable fam ily of m easu rab le s e t s { U j , form in g a b a s is fo r the B o re l s e t s , such that, given any two points x , y in a su itab le se t of m e a su re 1 , one can find a fin ite num ber o f s e t s Uit , U¡2, . . . , U¡ overlapp in g in chain ( i .e . v (Uij П Uij + 1) > 0) such that x e U ii and yeU ¡m (see F ig . 8).

F u rth e rm o re , to each of th ese s e t s U¡ the Hopf idea can be applied : in fac t, roughly speak in g , the s e t s U¡ can be thought of a s unions of p ie c e s of le a v e s of a con tractin g fo lia tion and, at the sam e tim e , a s unions of p ieces of le a v e s of an expanding fo liation ; fu rth erm o re , onezcan u se a " sy s te m of c o -o rd in a te s" b a se d on the h y p e rsu rfac e s C ^ n U ; = Cic and CEe П U¿ = C ie and the m e a su re o f a se t B cU ¡ can be com puted a s a double in teg ra l on the product o f the n atu ral m e a su re s dg- a, dj- cr induced by the R iem annian m e tr ic on C E o r C . . ic le6C se

M ore p re c ise ly , it i s p o ssib le to con struct two m e asu rab le partitio n s f c and Çe fo r each set Ui which a r e lo c a l contracting o r lo c a l expanding

200 GALLAVOTTI

le a v e s , and th ese two p artitio n s a re ab so lu te ly continuous with re sp e c t to each other ( se e below ).

The above d isc u ss io n should be understood a s an intuitive anticipation of the p re c ise defin itions to be in troduced in the next sectio n .

5. M EA SU RA BLE FO LIATIO NS

H ere we provide the p re c ise definitions needed to fully understand the sen ten ces o f the la s t section .

L e t (M, t , v) be a dynam ical sy stem .

D efinition 1 : a m e asu rab le se t U cM i s sa id to be m easu rab ly p a r titioned by a k-d im en sion al foliation g if:1 ) I i s a partitio n o f U.2) The e lem en ts С ^ Щ a re open k-d im en sion al p iecew ise sm ooth m anifolds

hom eom orphic to an open k -sp h e re 1.3) If v denotes the re s tr ic t io n of v to the su b a lgeb ra Yj (? ) of the B o re l

a lg e b ra in U co n sistin g of the m e asu rab le s e t s that a re unions of elem en ts o f § then

v(B) = J v ( d C E) J p ^ ( u ) d ? (j VB С UcU CgflB 1 6

w here pg- (u) > 0 a lm o st everyw here with re sp e c t to the natural m easu re(su rfac e m e a su re ) d ^ r on the m anifold.

fR em ark : the above double in te g ra l h as to be understood in the u su al

s e n se , i . e .

p ~ (u) d ~ (S E

c f Пв

m ust m ake se n se and m ust be an in tegrab le function with re sp e c t to v fo r a ll m easu rab le B c U .

D efinition 2 : If in U th ere a r e two m easu rab le p artitio n s f 1( g2, we s a y that g2 i s ab so lu tely continuous with re sp e c t to f j if ev ery elem ent o f | 2 in te r se c ts ev ery elem ent o f f i in ju st one point and if a ll We2H(?2 ) such that y(W ) > 0 a re such that Tg- (W nC{ > 0 fo r a ll ap art fo r a se t of

C ' s which a r e , how ever, contained in a null s e t . In th is c a se , we sayÇ 2<< ■

1 It is perhaps important to state explicitly what is meant by an open piecewise smooth k-dimensional manifold (see Ref. [1] ).

A closed k-dimensional smooth submanifold of a manifold M is a cf-smooth submanifold N homeomorphic to a closed k-sphere through a mapping ■which, in the neighbourhood of each point of N, is given in local co-ordinates by C2 functions having a limit on the boundary of N together with their derivatives; furthermore, if к > 2, the boundary 9N must consist of a finite number of closed smooth k-1 dimensional manifolds.

A closed submanifold is called a closed piecewise smooth submanifold if it consists of the union of a finite number of closed smooth submanifolds and if it is homeomorphic to a closed k-sphere.

An open piecewise smooth submanifold is a submanifold which is homeomorphic to an open к-sphere such that its closure is a closed piecewise smooth submanifold of the same dimensionality k.

I A E A - S M R - 1 1 / 2 5 201

R em ark : if « Ç2 arid f 2 «_Ç j , then it i s quite c le a r that, ap art fo r a "b a d " se t of lo ca l le a v e s Cfl and C fz contained in a null s e t^ th e in te rse c tio n s of any se t W of full m e asu re in U with C f i£ and C £ e Ç2 have fullm e a su re with re sp e c t to the n atu ral m e a su re s dg- cr and dg- cr, i . e .

Fi к

a ? (WnC ) = erg- (C ) fi 1 Fi

i f the C£ a re not "ex cep tio n a l" .si

D efinition 3 : Let (M, v, t ) be a dynam ical sy ste m and let f be a m e a su ra b le decom position of a m e asu rab le se t U cM . Then f i s sa id to be "co n trac tin g " i f fo r any two points x , у chosen a lm o st everyw here (with re sp e c t to the n atu ral m e a su re dg^cr) on a le a f C £ a re such that d(Tnx, т пу ) ----- *-0 with the p o ssib le exception of a se t of С 's contained in

a null s e t .S im ila r ly , we define an expanding m easu rab le partition , i . e . in the

above definition we re p la c e т by r'1.F ro m the above re m a rk s and definitions we re a l iz e that (M, t , v) is

going to be ergod ic if it i s p o ss ib le to con struct enough s e t s U adm itting m easu rab le fo lia tio n s of contracting and d ilating type which, fu rth erm ore , a re ab so lutely continuous with re sp e c t to each other.

6 . H EU RISTIC CONSTRUCTION OF THE FO LIATIO NS FO RBILLIA R D S

L et u s conclude by d isc u ss in g how one can attack the problem of finding the con tractin g and d ilating fo lia tion s in the c a se o f b i l l ia rd s .

We sh a ll only presen t h eu ristic argu m en ts.Suppose there is a con tractin g curve у through x e M . Then the

m appings T "1, T"2, . . . . m ust a ll be sm ooth on у (rem em b er that, by our conventions, T sends back into the p ast and T _1 sen ds into the future).

F ro m geom etric argu m en ts, it i s e a s ily shown that if cp = cp(r) i s theequation of a sm ooth curve Г and cp' = cp' ( r ') i s the equation of T Г, then

d<p' . J 1 1- f - т - - C O S cp' — --------------- г ---------------------------------dr' r \R c o sc p 1

Т (Г ’ У )+ "1 , l " ' d ^ co s cp R d r J

H ence, if T "1 i s sm ooth on у fo r i = 1, 2, . . . we find, by rep eated application o f the above form ula,

4^(x) = - c o s<p(x) (— ----— +-d rv \Rçoscp(x)t ( T _1x ) + ---- -

R c o s ^ T 'M т ( т Л ) + ------- 1

I____ +Rcoscp(T”zx)

- - k c ( x ) C O S ip(x)

202 GALLAVOTTI

w here кс (x) i s the function defined by the continued fraction in side the p aren th e se s ; it con v erges sin ce |т (Т " ‘ х ) |ё ro > 0 b e cau se Q¡nQj = ф, i f j .

It i s a lso not d ifficu lt to se e from the above equation that the curve 7 (if ex istin g) m ust be such that the d istan ce betw een the T "1 im ag e s of any two points on 7 tends to zero a s i -» cc ( i . e . 7 i s actu ally a contracting curve).

T he f i r s t r e a l problem is to show that the above d iffe ren tia l equation actu ally h as a solution; th is se e m s to be a d ifficult problem sin ce it is e a s ily re a liz e d that k c ( x ) is discontinuous over a dense se t and, on it s set o f continuity, it i s not at a l l n icely behaved.

H ow ever, it i s p o ss ib le to prove that if the d istance of T " 'x from the sin gu la rity points of T does not tend to zero too fa st a s i -* « , then the equation for 7 has a solution in a neighbourhood of x.

A s im ila r construction p rov ides the p iece s of the expanding le a v e s .The se t of the x in M fo r which T _1x does not get "to o " c lo se to the

sin gu la rity lin e s o f T can be shown to have m e a su re 1, so that the d ifferen tia l equation defining the contracting le a v e s h as a solution for a se t of in itial data having full m easu re [ 3 ] .

At th is point one h as to con struct a fam ily { U j } of m easu rab le s e t s which a re m easu rab ly decom posab le by expanding and contracting fo lia tio n s.

The con struction s and p roofs re la te d to th is point are a very nice p iece o f geom etry and re a lly at the heart o f the th eorem . They a re ra th er s im ila r to the analogous con struction s encountered in the theory of the C -sy ste m s [2 , 3] . H ow ever, let us note the b a s ic conceptual d ifferen ce between C -sy s te m s and b ill ia rd s : fo r С - sy s te m s , each point x belon gs to contracting and d ilating le a v e s of foliation and, fu rth erm o re , the dependence of the lea f upon the point x i s sm ooth . In the b ill ia rd s c a se , we have a situation in which the le a v e s o f both the contracting and dilating fo lia tio n s cover M only a lm o st everyw here; fu rth erm ore , they typ ically end on sin gu larity l in e s o f T o r T "1 o r it e r a te s o f them and, i f y ( x ) i s a le a f through x, its dependence on x i s fa r from being sm ooth.

The in te re ste d re a d e r i s r e fe r re d to the fam ous p ap ers [1 -3 ] .

R E F E R E N C E S

[ 1 ] SINAI, Y a, D yn am ica l system s w ith co u n tab le Lebesgue spectrum , II: Am . M ath. Soc. T rans.2 (1968) 34.

[ 2 ] ANOSOV, D. , SINAI, Y a. , Som e sm ooth ergod ic system s, Russ. M ath. Surv. : 22 (1967) 103.[3 ] SINAI, Y a. , System s w ith e la s tic re flec tio n s , Russ. M ath. Surv. 25 (1970) 137.

IAEA-SMR-11/26

SOME REMARKS ON QUASI-ABELIAN MANIFOLDS

F. GHERARDELLI, A. ANDREOTTI Istituto M atem atico,U niversité di Firenze,F lorence, Italy andIstituto M atem atico,Universitá di Pisa,Pisa, Italy

Abstract

SOME REMARKS ON QUASI-ABELIAN MANIFOLDS.Som e results ob ta ined in th e study o f quotients of C n by d isc re te sub-groups Г o f n o n -m a x im a l

rank are presented .

In th is p ap er, we sh all p re sen t som e re su lt s obtained by the auth ors on studying quotients of (Cn by d isc re te su b -gro u p s Г of non -m axim al rank.The r e su lt s we p re se n t a r e v ery sim p le and have been obtained by e lem en tary m ethods.

1. C o n sid er a com plex L ie group G without non-constant holom orphic functions. Then it i s w ell known that

a) G is A belian and of the fo rm (En/ r n+m w here Г i s a d isc re te (Abelian) sub-group of Œn of rank n + m , n < n + m < 2n such that Г gen era te s Œn over Œ and Г ® R s IR£+m.

b) If G and G' a re two such L ie grou ps and f: G ->G ' i s a holom orphic m ap such that f (0) = 0 then f i s a hom om orphism of groups.

c) The r e a l sp ace IR"+m contains a com plex su b -sp a ce IF of com plex d im ension m (and not g re a te r ).

d) Of c o u rse , not a l l the quotients ŒN/ r M(N + 1 s M S 2N) a re without non-constant holom orphic functions; but it i s e a sy to se e that ev ery quotient C N/ r Mis a C a r te s ia n product of fa c to r s of the follow ing types: <E, С * , T , <En/ r n+m where T i s a com plex to ru s and (Пп/ Г п+Ш,0 < m < n, i s a non-com pact A belian L ie group without non-constant holom orphic functions.

e) F ro m the r e a l point of view

( C n / r n + m = ^ r n + m x R n -m

w here i s a r e a l to ru s of r e a l d im ension n + m in which F h as a dense im age.

The study of the quotients <Еп/Г п +т(0 < m < n) w as in itiated by C ousin ([2, 3]), who, in p a r t ic u la r , studied the c a se n = 2, m = 1. R ecently , the study of th ese quotients a s non-com pact A belian L ie grou ps w as aga in taken up by M orim oto in R e fs [4, 5], w here a l l the abovem entioned p ro p e rtie s a r e derived.

203

204 GHERARDELLI and ANDREOTTI

2. In the study of com plex to r i, it i s a c la s s ic a l problem to a sk fo r the conditions fo r a com plex to ru s to be an a lg e b ra ic m anifold, i. e. an A belian m anifold. In analogy, we want conditions fo r X = Œn/ r n+m,0 < m < n, to be a lg e b ra ic q u a s i-p ro je c tiv e , i. e. a Z a r isk i open se t of som e p ro jectiv e a lg e b ra ic v arie ty . L e t 7r:(Cn -»X the natural p ro jectio n ; follow ing the c la s s ic a l approach , in the c a se of com plex to r i ([1 , 9.1) we want conditions fo r the ex isten ce on of a non-degenerate m erom orp h ic function f, such that 7 r*f can be rep re sen ted a s a quotient of two en tire functions having the p ro p e rtie s of theta functions with re sp e c t to Г .

Fo llow in g the c la s s ic a l p roced u re step by step , we obtain the follow ing n e c e s sa r y conditions fo r the ex isten ce of f:

A) T h ere e x is t s a H erm itian form ^ fo n Œn X (Cn such that ^ fxt p o sitiv e and

B) ^ ¡ r x r ha s in tegra l v a lu es.L et С be the m a tr ix of a b a s is of Г , then, in m atr ix fo rm , condition B)

can be w ritten

‘С H С - *С ‘Н С = 2 i A (1 )

w here H i s a m atr ix of the H erm itian fo rm ,gând A is a sk ew -sy m m etric sq u are m atr ix of o rd e r n + m with in teg ra l en trie s.

R e m ark s: 1. If m =n, conditions A) and B) a re the c la s s ic a l Riem ann re la tio n s.2. A) and B) a r e fu llfilled if X = Œn/ F n+mh as a Hodge m e tr ic , i. e. a K áh lerian m e tr ic such that the a sso c ia te d tw o-form h as in tegra l p erio d s.

D efinition 1: If conditions A), B) hold, we say that X i s a q u asi-A b elian m anifold. F o r a q u asi-A b e lian m anifold we have the follow ing

T h eorem 1: L e t X be a q u asi-A b e lian m anifold; than X is a coverin g of an A belian m anifold.

C o ro llary : The ex isten ce of a Hodge m e tr ic on X is equivalent to conditionsA ), B) and a lso to the ex isten ce of a non-degenerate m erom orph ic function on X of the p rev io u s type.

Sketch of the proof. The H erm itian form is determ ined by conditions A) and B) only up to a H erm itian form A sy m m e tr ic on Г X Г . It i s then an e x e rc ise in lin e a r a lg e b ra to show that there e x is t s a v ecto r 7 G(Cn such that i) The group 1} gen erated by Г and у h a s a rank n + m + 1 over IR,

ii) F j ®IR contains a com plex su b -sp a ce Tj of d im ension m + 1 , iii) 9" can be chosen in such a way that ^f+^¡rlxFl i s positive and

Im [gtf + S?1) h a s in teg ra l v a lu e s on Tj X Гг. R epeating th is argum ent n - m tim e s we obtain the theorem .

3. The rank of the sk ew -sy m m etric in tegra l m atr ix A in Eq. (1) i s an even num ber not l e s s than 2m : 2m S rank A S n + m .

D efinition 2: The q u asi-A b elian m anifold X = (Dn/ r n+mis called of kind p if rank A = 2m + 2p (0 s 2 p s n - m ) .

IAEA-SMR-11/26 205

T heorem 2: L e t X = Œn/ E n+m be a q u asi-A b e lian m anifold of kind p. Then X i s a fib re bundle over an A belian m anifold Y of d im ension m + p with f ib r e s (EPX ((C*)n"m"2P ( 0 S 2 p S n - m ) . In other w ords: E v e ry q u asi-A b elian m anifold i s an A belian extension of the lin ear group (EPX( <ц *)п" ш _2Р b y a n

A belian m anifold. T h is p ro v es that our q u asi-A b elian m an ifo lds a re e s se n t ia lly those con sid ered under th is nam e by S e v e r i [ 8]. T h e ir study a s extension of a lg e b ra ic grou ps can be found in S e r re [7] where a lso other re fe re n c e s a r e given.

4. T heorem 1 and 2 a re obvious in the follow ing exam ple . L et X = (Е2/ Г 3; h ere conditions A) and B) a r e a lw ays fu lfilled . B y lin e a r tran sfo rm atio n s in Œ2 the m atr ix С of Г can be reduced to the form

Then we have:

X i s without non-constant holom orphic functions if and only if the n um bers 1 ,a,f¡ a re lin e a r ly independent over the ra tio n a le . The r e a l sp ace Г ® IR = R 3 contains a com plex line F on which, at m o st, one g en era to r of Г i s lying. Suppose, fo r sim p lic ity , that Г does not contain any gen era to r of Г and a lso that Im a > 0. Then the m atr ix С (o r the group Г) can be com pleted

with another colum n v ecto r ^ ^ s u c h that

is a R iem ann m atr ix , and th is i s the content of T heorem 1.L e t now Uj, cj2, u3 be a b a s is of Г; p ro jec t Œ2 over F in the d irection

of the com plex line defined by щ . Then the p ro jec tio n s of u2 and 103 define a la tt ice 7 on F . P a s s in g to the quotient (Еп/ Г we obtain a p ro jectio n of X onto the e llip tic curve F /7 with f ib r e s Œ* (a s stated in T heorem 2).

w here a, ¡3 a re two com plex n um bers so that

is sy m m etric and

206 GHERARDELLI and ANDREOTTI

R em ark that the p rev iou s construction can be done in an infinite num ber of w ays: we can s ta r t fro m any b a s is of Г . B y d irec t com putation it is e a sy to show that the e llip tic cu rv e s which we obtain by th ese p ro jectio n s a r e , in g en era l, n on-isom orph ic so that (С2/ Г 3 i s fib red over infintely m any e llip tic cu rv e s and we have a sim p le exam ple of a (non-com pact) com plex m anifold with in finitely m any a lg e b ra ic s tru c tu re s underlying the sam e com plex stru c tu re (co m p are R ef. [ 6], note p. 34).

R E F E R E N C E S

[1 ] CONFORTO, F . , A belsche Funktionen und a lg eb ra isch e G eo m e trie , Springer, B erlin (1958).[2 ] COUSIN, P . , Sur les fonctions périod iques, Ann. S c. Ec. N orm . S u p ., Paris 19 (1902).[3 ] COUSIN, P . , Sur le s fonctions périodiques de deux v ariab les , A cta M at. 33 (1910),[4 ] MORIMOTO, A . , Non co m p ac t com p lex Lie groups w ithout non constan t ho lom orph ic functions, Conf.

in C om plex A nalysis, M inneapolis (1966),[ 5 ] MORIMOTO, A ., O n c la ss if ic a tio n of non c o m p ac t com p lex ab e lian Lie groups. Trans. A m er. M at.

Soc. 123 (1966).[ 6 ] MUMFORD, D , , A b e lian v a rie tie s , Oxford (1970).[7 ] SERRE, J. P . , G roupes a lgébriques e t corps de classes. Paris, H erm ann (1958),[8 ] SEVERI, F . , F unzioni q u a s i-a b e lia n e . Pont. A c. Sc. (1947).[9 ] SIEGEL, C . L . , A n a ly tic functions o f severa l com p lex variab les. I. A .S . (1948).

[1 0 ] WEIL, A . , In troduc tion à l 'é tu d e des varié tés kah lé rien n es. P aris, H erm ann (1958).

IAEA-SMR-11/27

LIFE AND DEATH OF THE BERNSTEIN PROBLEM

E. GIUSTIDepartment o f M athem atics,Stanford University,Stanford, C a l i f . ,U nited States o f A m erica

Abstract

LIFE AND DEATH OF THE BERNSTEIN PROBLEM.T h e Bernstein p rob lem as w e ll as som e o f its proofs and ex tensions are p resen ted .

1 . In 1915, B e rn ste in [2 ] proved h is ce leb ra ted re su lt concerning en tire m in im al g rap h s:

B e rn ste in 's T h eo rem . Let u (x ,y ) be a C2 function in IR2, solution to the m in im a l- su r fa c e equation:

u' = о (1)

эх 4 ^ 2^ у эу

in a ll o f ]R2. Then the graph of u i s a p lane.S in ce the ap p earan ce o f B e rn s te in 's p ap er, se v e ra l proo fs of the

th eorem have been published; the follow ing one, due to N itsch e [ l i j á i s one of the m ost elegant:

L e t us o b se rv e f i r s t , with H einz, that B e rn s te in 's r e su lt follow s from a th eo rem , o rig in a lly proved by Jo r g e n s , sta tin g that if Ф (х ,у ) i s a function v erify in g the equation

Ф Ф - Ф2 = 1 (2)XX yy x y ' >

in IR2, then Ф m ust be a polynom ial o f the second d egree .The link betw een th is re su lt and B e rn s te in 's th eo rem l ie s in the fact

that the m in im a l- su r fa c e equation, which can be w ritten a s

(1 + u2)uxx - 2uxuyuxy + (1 +u2)uyy = 0 (3)

i s the n e c e s sa r y and su ffic ien t condition fo r the ex isten ce o f a function Ф (х ,у ) such that

1 + u‘

■J 1 + uУ

Ux Uyфху= — .... . (4)

V l + u 2+ u 21 + uj

b y - г -----¿----2* / l + u 2 + u2 x у

207

208 GIUS TI

Such a function Ф i s e a s i ly seen to v e r ify the hypothesis o f Jô r g e n 's theorem , and hence to be a polynom ial o f the second d eg ree . T h is im p lie s , in turn, that the d e riv a tiv e s of u a re con stan ts.

We com e now to N itsch e ’ s proof of Jô r g e n 's th eo rem . We can suppose that Ф(х, y) i s a convex function, so that the m ap

J 5 = x + Фх(х ,у )

l Г) = у + Фу(х ,у )

i s a d iffeom orph ism of IR2 onto it s e lf .If we put Ç = I + Í17, and

w ( 0 = x - Фх(х, y) - i(y - Фу(х ,у ))

(here x and у have to be understood as functions of f and rj), the function w(f ) i s an en tire holom orphic function. In addition, we have

1 - |w '(H |2 2 + Ф + Ф* XX * y y

from which follow s that w '(£) i s bounded, and hence constant by the L iouville th eorem . On the other hand, we have

I 1 - W' I2Ф - •

1 - |w- I2

= _[l± w ^yy I 12 1 - I w' I

so that Ф i s a polynom ial o f degree tw o.

2. It i s n atu ral to look fo r an extension of the B e rn ste in th eorem to d im ensions higher than two, i . e . to ask whether o r not an en tire solution to the m in im a l- su r fa c e equation in H n

U*i[ s ; ( 7 = s ) - " <s>1 + Du

(I Du I denotes the length of the vecto r g rad u) m ust n e c e s sa r i ly be lin e a r .On the other hand, m ost o f the different p roo fs of B e rn ste in 's theorem

a r e b a sed on com plex function theory and hence cannot be extended to m ore than two d im en sio n s.

It w as only in 1962 that F lem in g [ 8 ] found a new proof o f the theorem , usin g a m ethod independent of the num ber of d im en sion s, and opening the way fo r fu rth er developm en ts.

3 . T o sketch F le m in g 's id ea , le t u s d e sc r ib e b r ie fly De G io rg i 's fo rm a lism fo r m in im al su r fa c e s .

IAEA-SMR-11/27 209

L et A be a B o re l se t in ]Rn+1 and le t ipA be it s c h a ra c te r is t ic function. We say that the boundary of А , ЭА, i s an orien ted h y p ersu rface (brie fly : a su r fa c e ) i f the d e r iv a tiv e s of ¡pA, in the sen se of d istr ib u tio n s, a re Radon m e a su r e s . In th is c a s e , fo r every B o re l se t K , we define the a r e a of the p art o f the su r fa c e ЭА ly ing in К a s the to tal v a ria tio n of the v ec to r-v a lu ed m e a su re D <pA in K:

The re a so n for the above definition w ill becom e apparent if one c o n sid e rs a se t A with a sm ooth boundary. Let g(x) be a sm ooth v ec to r-v a lu ed function with com pact support; we ob tain .from the G a u ss-G re e n form ula:

w here v i s the ou ter n o rm al v ec to r to ЭА and dHn i s the su r fa c e m e a su re .

On the other hand, we have

so that, for re g u la r s e t s , our definition co in cid es with the u su al su r fa c e m e a s u r e .

The follow ing definition o f m in im al su r fa c e s i s now quite n atu ral:

D efinition. Let fi be an open se t in ]Rn + 1. We say that ЭА i s a su r fa c e of le a s t a r e a , with re sp e c t to fi, if for every se t B , which co in cides with A except p o ssib ly in som e com pact se t K c f i , we have

If ЗА i s a su rfac e of le a s t a r e a , we sh a ll c a ll A a m in im al se t . Let us re m a rk that the re a so n fo r in troducing the com pact se t К in the definition i s that a su r fa c e o f le a s t a r e a m ay have infinite a r e a (like, e .g . a h y p erplane) .

T he re la tio n betw een m in im al s e t s and so lu tion s to the m in im al su r fa c e equation l ie s in the fac t that a C2 function u(x) in IRn i s a solution to E q . (6 ) i f and only if the set

к

A За

/div g dx = / <pA div g dx = - < D tpA, g >/A

w here У i s the d istribu tion b rack e t.It follow s at once that

D<pA = - v dH,n I ЭА (7)

and hence

(8 )

D <pA| (K) S I D цс0 j (K)

(9)

has the boundary of l e a s t a r e a in IRn+1.

210 GIUSTI

4 . The idea of F lem in g i s the follow ing: Let t > 0, let A be the se t (9) and let

At = | z e H n+1 : t z e A

It i s c le a r that 9At i s a su r fa c e o f le a s t a r e a in IRn+1, fo r ev ery t > 0. We can choose a sequence { t n}, tn oo, such that the sequence <рд con v erges in L loc ( K n+1) to the c h a ra c te r is t ic function tpc of som e se t C . n

The se t С i s a cone and, what i s c ru c ia l , i f ЭС i s n o n -sin gu lar ( i .e . if ЭС i s a hyperplane), then ЭА it s e l f i s a hyperplane.

In th is way, the extension of B e rn ste in 's theorem to IRn i s reduced to the prob lem of the ex isten ce o f s in g u la r m in im al cones in IRn+1. S ince no such cone can e x is t in ]R3, F le m in g 's argum ent g iv es a new proof of the B e rn ste in th eo rem .

5 . The next step w as taken by De G iorg i [ 7] . He showed that the lim it cone С cannot be s in g u la r only at the v e r te x . In fac t, if С i s s in gu la r , then it m ust be a v e r t ic a l cy linder: С = С' X R o v er a m in im al cone С ' С K n.

T h is extends B e rn ste in 's th eorem to g rap h s in R 3, s in ce now the nonex isten ce of m in im al cones in IRn im p lie s the B e rn ste in th eorem for m inim al g raph s o v er IRn.

6 . The im portan ce of m inim al cones cannot be fully understood without m entioning the problem of the re g u la r ity o f su r fa c e s o f le a s t a re a . Let T2be an open se t in IRn and let A be a m in im al se t in Г2. We can suppose that 0 ЕЭАПГ2. Let us co n sid er, a s b e fo re , the se t s

At = x e E n : t x e A

We can now let t -*■ 0 through a sequence tn in such a way that <pA con verges

lo ca lly to cpc. The se t С i s a m in im al cone; roughly speak in g , it i s the tangent cone to ЭА at 0 .

It is p o ssib le to show that if the cone С i s a h a lf-sp ac e then the su rfac e ЗА i s re g u la r in a neighbourhood of the o rig in .

7. In 1966, A lm gren [1 ] p roved the non -existen ce of s in gu la r m inim al cones in IR4, and in 1968 S im ons [12] extended th is re su lt up to IR7.

The idea o f S im ons c o n s is ts in evaluating the e igen valu es o f the b ilin ear form which g iv e s the second v aria tio n o f the a r e a fo r a sta tio n ary cone. A c a re fu l e stim ate show s that, fo r n s 7, the f i r s t eigenvalue of th is form is n egative , and hence p ro v es the ex isten ce o f v a r ia t io n s which d e c re a se the a r e a . In the sa m e p ap er S im ons g iv es the exam ple o f the cone

r< - f td 8 2 , 2, 2 , 2 . 2, 2 , 2, 2]C - j x e l R : x 1+ x 2+ x 3+ x 4< x |.+ x¡1 + x, + x 8j

which i s lo ca lly stab le , in the se n se that every com pact v a ria tio n of С in itia lly in c r e a se s the a r e a .

T h is cone i s actu ally a m inim al cone (in the se n se o f our definition), a s shown by B o m b ieri, De G iorg i and G iusti [ 3 ] , and prov id es a counterexam ple to the problem of in te r io r re g u la r ity o f m in im al s u r f a c e s .

With the help of the cone С and of som e ca lcu la tio n s, the sam e authors w ere ab le to con struct com plete m in im al g raph s ov er IRn, fo r n è 8, d ifferen t from h yp erp lan es.

IAEA-SMR-11/27 211

8 . We can su m m arize the p rev iou s d iscu ssio n a s fo llow s:

T h eorem A . Let u(x) be a C2 function in IRn , solution to the m in im a l- su r fac e equation

}Then, if n S 7, the graph o f u i s a h yperplane.

T h is is not the c a se fo r n § 8 .

9 . The p reced in g th eorem g iv e s a com plete solution to the B ern ste in p rob lem in IRn, but sin ce the an sw er i s negative fo r h igher d im en sion s, n atu ra lly , som e additional question s a r i s e .

A f ir s t kind o f problem i s the se a rc h fo r additional conditions on u(x) that im p ly , o f c o u rse , fo r a num ber of d im ensions h igher than seven , that the graph of u i s a p lane. We sta te h ere two re su lt s in th is d irection :

a) M o ser [ 1 0 ] . If, in addition to the a ssu m p tio n s of T h eorem A , we su ppose that the d e riv a tiv e s o f u are bounded, then u i s l in e a r .

b) B o m b ieri, De G iorg i and M iranda [ 4 ] . If the function u(x) s a t i s f ie s the inequality

for the grad ien t o f positive so lu tions to the m in im a l- su r fa c e equation in the b a ll o f rad iu s R . We rem ark that the e stim ate (10) g u aran tees the re g u la r ity of g en era lized n o n -p aram etric m in im al su r fa c e s (se e , e .g . R ef. [9 ] ) .

10. Another type o f question con cern s the behaviour of com plete m in im al g raph s at infinity. V ery little i s known on th is su b ject; we re fe r h ere to a paper of B o m b ieri and G iusti [ 5] .

L e t S be the su r fa c e graph o f u, le t O eS and let Sr = S n B r be the part o f S ly ing in the b a ll o f rad iu s r . Iî0is the L ap la ce o p era to r on S , we have the follow ing re su lt :

L e t g(x) be a p o sitiv e su perh arm on ic function on S ( i .e . ® g s 0). Then

u(x) È - к (1 + I x I )

then u i s l in e a r .The proof o f re su lt b) depends on the a-p r io r i e stim ate :

(10 )

i |2 -1/2In p a r t ic u la r , s in ce i/(x) = (1+ |Du| ) i s su p erh arm on ic , we e a s ily obtain

1 + I Du I dx (И )

212 GIUS TI

w here я, i s the p ro jectio n of Sr on IRn:

| x e E n: Ix 12 + |u(x) |2 < r 2{ }F ro m (11) one can derive the e stim ate

ln(12)

In fac t, le t p > 0 and let r 2 = p2 + sup | u(x) |2.I x | < p

The b a ll of rad iu s p in K n i s contained in 7rr, and hence

m eas v È u pn r n

On the other hand,

и r n + 1

n

and inequality (12 ) fo llow s at once.

Inequality (12) i s , in som e se n se , an im provem ent of inequality (10); let us point out exp lic itly that it i s only v a lid fo r com plete m in im al g rap h s.

We re m ark that the inequality

and hence a growth condition on com plete m inim al g rap h s.Although n does not seem to be the b est exponent and som e im provem ent

can p o ssib ly be obtained, the way from inequality (12) to inequality (13) se e m s to be quite long.

[1 ] ALMGREN, F. J. , J r . , Som e in te rio r reg u la r ity theorem s for m in im a l surfaces and an ex tension o f B ernstein 's th e o rem , Ann. M ath. _85 (1966) 277.

[2 ] BERNSTEIN, N .S . , Sur un th éo rèm e de g éo m é trie e t ses app lica tions aux équa tions aux dérivésp a rtie lle s du type e l l ip t iq u e , COmm. Soc. M ath, de Kharkov (2èm e sér. ) (1915-1917), 38. (SeeM ath . Z e it. 26 (1927) 551, for a G erm an tra n s la tio n .)

[3 ] BOMBIERI, E. , DE GIORGI, E. , GIUSTI, E. , M in im al cones and the Bernstein p ro b lem , Inv. M ath. 1_

(1969) 243.[4 ] BOMBIERI, E. , DE GIORGI, E . , MIRANDA, M. , Una m agg io razione a p rio ri re la tiv a a ile ipe rsuperfic i

m in im a li non p a ra m e tr ic h e . A rch. Rat. M ech. A nal. ^ 2 (1969) 255.[ 5 ] BOMBIERI, E . , GIUSTIv E . , A H arnack 's type in e q u a lity for e l l ip t ic equa tions on m in im a l surfaces,

Inven t. M ath. JL^(1972) 24.[6 ] DE GIORGI, E. , F rontiere o r ie n ta te d i m isura m in im a , S em inario M atem ático S cuola N orm , Sup. Pisa

(1960-1961). E d itrice T é cn ico S c ie n tif ic a Pisa (1961).

supI x | < p

(with exponent 1 on the righ t-han d sid e ) would im ply that

|u (x)| S k ( l + |x | ) 4

R E F E R E N C E S

IAEA-SMR-11/27 213

[7 ] DE GIORGI, E. , U na estensione d e l te o rem a di B ernstein, A nn. S cuola N orm . Sup. Pisa (3) 19 (1965) 79.[8 ] FLEMING, W .H ., On th e o rie n ted P la teau p ro b le m , Rend. C irc . M at. P alerm o (2) 11 (1962) 69.[9 ] GIUSTI, E. , S uperfic i c a rte s ian e d i a rea m in im a , Rend. S em . M at. Fisico M ilano 40 (1970) 1.

[1 0 ] MOSER, J. , On H arnack 's theo rem for e l l ip t ic d iffe re n tia l eq u a tio n s , C om m . Pure A ppl. M ath. 14 (1961) 577.

[1 1 ] NITSCHE, J . C .C . , E lem en tary p roof o f B ernstein 's theo rem on m in im a l surfaces, Ann. M ath. 66_(1957) 543.

[1 2 ] SIMONS, J. , M in im al varie ties in R iem annian m anifo lds, A nn. M ath. (2) 88 (1968) 62.

IAEA -SMR-11/28

INVARIANTS OF FOLIATIONS

C. GODBILLONDépartement de m athém atique,Université de Strasbourg,France

Abstract

INVARIANTS OF FOLIATIONS.T h e defin itio n o f a fo lia tio n is g iv e n . An in v a rian t o f fo lia tions of cod im ension one is in troduced and

re la tio n s w ith the G elfand-Fuks cohom ology a re estab lish ed .

INVARIANTS OF FOLIATIONS

We give h ere som e elem en tary id ea s on the new cohom ological in v arian ts of fo lia tion s introduced in Ref. [4] . F o r fu rth er developm en ts, se e R e fs [1, 2 ].

1. F o lia tio n s [5]

C on sid er f i r s t a d ifferen tiab le m anifold M of d im ension m , and a sm ooth v ecto r field X on M without s in g u la r it ie s . E ach m ax im al in tegra l curve of X is then a connected on e-d im en sion al subm anifold of M , and thefam ily (0 £) of th ese o rb its has the follow ing p ro p e rtie s :

i) M = U 0 { ;ii) Oc П o j = ф fo r t j;

iii) if EXCTX(M) is the tangent sp ace at x to the orb it th ro u gh x (E x is thesu b sp ace of TX(M) gen erated by X (x)), then E = и E x is a sm oothsubbundle of rank one of T(M ). xeM

Such a situation is a fo liation of d im ension one of M. M ore gen era lly :

D efinition. A fo liation & of d im ension q (or of codim ension m -q) of M is a fam ily (F{ ) of connected subm anifo lds of d im ension q of M having the follow ing p ro p e rtie s :

i) M = U{ F{ ;ii) F{ П Fj = ф fo r ¿ ф j;

iii) if E x с TX(M) is the tangent sp ace at x to the subm anifold F{containing x , then E = и E x is a sm ooth subbundle of rank q of T(M ). xeM

E ach Fg i s a le a f of ,9^ and E is the tangent subbundle of ¡P~.

E x am p le s . If M = Q X N, the fam ily (Q X {n l ) neN is (if Q is connected) a fo liation of M.

M ore g en era lly , if p : M ->• N is a su b m ersio n of Mm into Nn, m s n, the fam ily of the connected com ponents of the in v e rse im ag es p'Mx), x e N , is a fo liation of codim ension n of M.

215

216 GODBILLON

If & i s a fo liation of codim ension one of M its tangent subbundle E can be (locally ) defined by a P faffian form (i. e. a d ifferen tiab le form of d egree one) со without s in g u la r it ie s : E x = {e G TX(M)| <e ,u (x )> = 0 } . Since u induces the ze ro form on each le a f F of the sam e holds fo r du ; and, th ere fo re , du is d iv isib le by u . So uAdu.= 0.

C o n v erse ly , let u be a P fa ffian form without s in g u la r it ie s on M. An in teg ra l subm anifold of u is a sm ooth connected subm anifold of codim ension one of M on which u is ze ro . Then one h as:

F rob en iu s th eo rem . If u A d u = 0, there e x is t s fo r each point x e M an in te g ra l subm anifold of и containing x.

M o reov er, if N j and N2 a re two such in teg ra l su bm an ifo lds, there e x is t s an in te g ra l subm anifold P of u such that (in som e sen se ) P D N j U N2.

T h is allow s us to introduce the notion of "m a x im a l" in te g ra l subm anifold of u ; and the fam ily of these subm anifolds is a foliation of codim ension one of M whose tangent subbundle is (globally) defined by u . One sa y s that u i s an in tegrab le P fa ffian form .

E x a m p le s : i) If u is a c lo sed P faffian form without s in g u la r it ie s (du = 0)on M , u is in tegrab le . M o reov er, if M is com pact a l l the le av e s of the co rresp on d in g foliation a re d iffeom orphic [5 ].

ii) Let N be a m anifold with boundary, and f be a d ifferen tiab le function on N having 0 a s a re g u la r value and such that f _1(0) = 3N. Then u = d f+ fd0 d efin es a fo liation of codim ension one of M = N X S 1 having the boundary 3M = 3N X S 1 a s a union of le a v e s .

iii) T aking N = D2 (the tw o-d im ensional d isk ) in the p reced in g exam ple one g e ts a fo liation of codim ension one of the to ru s D2 X S 1. Then, gluing two such fo lia ted to ru se s by identifying the m erid ia n s of the boundary of one with the p a r a l le ls of the boundary of the o ther, one obtains a foliation of codim ension one of the three sp h ere S3: the Reeb fo liation of S 3.

2. An in varian t of fo lia tio n s of codim ension one [4]

Let S ^ b e a foliation of codim ension one of M defined by a P faffian form u, S ince u A d u = 0 one can w rite du = u A u j. By d ifferen tia tin g one ge ts 0 = u A d u j ; so that du j i s , too, d iv isib le by u : d u j = uAu2.

C on sid er then the th ree-fo rm Г2 = -UjAdUj = UAU|AU2. It is c lo sed (dfi= 0); and thus it d efin es a cohom ology c la s s [ Q] in the de Rham cohom ology group H3(M ,R ).

P ro po sition 1. The cohom ology c la s s [Г2] depends only on

P ro o f i) If du = u i lu j one h as uA(ux - иг') = 0; and, th ere fo re ,u •[ = u j+ f u dUj' = dUj + df Au + fdu

- Uj'AdUj1 = - UjA du j - u-l A df A u = Q - d(f du).

ii) If u " i s another P faffian form defining , one has u " = g u , w here g i s a d ifferen tiab le function on M without zero . Then du" = u "A u j' where u" = Uj - d g /g , and

dir i i- U j'A dU j = -U jA dUj + A dw = П + d (L og | g | dUj ) q. e. d.

IAEA-SMR-11/28 217

E x a m p le s , i) If и is c lo sed [Í2] = 0.

ii) If u = df + fdS, a s in exam ple ii) of sectio n 1, u = dô and [Г2] = 0.

iii) L et G = S L (2 , R) = {(:a b .с d.;) ad - be = 1 f.}■

T h ere is a b a s is a , о?}, or2 of the sp ace of left in varian t P faffian fo rm s on G such that

The form a is thus in tegrab le and defines the fo liation <3 whose le av e s a re the left c o se ts of the subgroup

and A = aA a1A a 2 is a volum e form on G.T h ere e x is t s a c lo sed d isc re te subgroup Г of G such that the quotient

m anifold M = F \ G is com pact. The fo liation Ç& of G being left invariant induces a foliation 5 ^ of M fo r which Í2 i s a volum e fo rm ; and, th ere fo re , [Г2] =j= 0 (this im portant exam ple is due to R o u ssa r ie ) .

iv) T hurston has shown that on S 3 the invariant [f2] can be any cohomo' c la s s in H3(S3, R) [6] .

3. R elation s with the cohom ology of G elfan d-Fuks [3 ].

D ifferen tiatin g the re la tio n d u 1= шЛи2 one obtains 0= u A ^ A U g - du2 ), so that du2 = U jA u 2 + u A u 3,

It is p o ss ib le to ite ra te th is p roced u re and introduce P fa ffian fo rm s u)k , k G N. T h ese fo rm s a r e not uniquely determ ined , but one has

da = of A

dffj = a A a 2

d a 2 = ° 1Л а 2

1. J

where the co e ffic ien ts c^ . a re "u n iv e r sa l" :i, J

fo r i+ j ^ k+1

fo r i+ j = к + 1

2 1 8 GODBILLON

L e tл/Ъе the r e a l L ie a lg e b ra with b a s is {Y 0, Y1 ( . . . , Yk , . . and L ie b rack et

[Y. , Y. ] = V ck Y = i (l + j ~ 1 ) '~ Y1 ■> L i.J к 2 i t -, > + J

b * J *i + i - l

If X k = 2k; Yk one has [X ¿,X j] = ( j- i) X i+j_1 ; so that is iso m o rph ic tothe L ie a lg e b ra of fo rm a l vecto r fie ld s in one v a r iab le :

jrf = { x = V a xk -^-T, У a. x 1 , У b. x J = У (j- i) a.b. x*+) 1 -Д-l z_, к 9xJ> _Z_, 1 Эх L—i J 9x_ z_, 1 J 8x

L e t j^ 'b e the sp ace of continuous lin ear fo rm s on (i. e. fo rm s which depend of a fin ite num ber of coeffic ien ts a k):

has a s a b a s is the fo rm s

:Iaixi4 ( - i r k ' . a„

Denote by C* (jzt) = EC (j<0 the ex te r io r a lg e b ra of (sp ace of continuous cochains on já), and by d the u su a l coboundary o p era to r in C * ( jrf) given by

<MZ 1......... z P+i) = X (- 1 )i+i “ ( Ц , z. ] , z , . . , z . , . . , z . , . . , Zp+1)i< j

One h as d°d = 0; and the quotient sp ace H* (,c/) = K erd /Im d is the continuous cohom ology sp a c e of the L ie a lg e b ra ji (Gelfand- Fuks cohom ology of jd).

The L ie a lg e b ra o p era te s on Qr{j>i) by the L ie derivation L x = ix d + d iJ{ (where ix is the in terio r product); and one has

v (l k >4 w h e r e K = 4

d a = ) ck a A a к 1_л i. ) ‘ J

i, i

So:

P ro po sition 2. T h ere e x is t s a m orph ism of com plexes h from C*(,rf) to the de Rham com plex Л(М) of M such that h(aQ ) = u.

T h is induces a m orph ism h * from the G elfan d-Fuks cohom ology H*(jd)to the de Rham cohom ology H *(M , R); and the c la s s M is in the im age of h *.

In fac t:

P ro po sition 3 [3 ] . One has Hk(jj/) = 0 fo r к =^0,3H3U ) = R with gen erato r the c la s s o f a 0Ao1 A^2.

IAEA-SMR-11/28 219

P ro of. L et H and К denote the fo rm a l v ec to r fie ld s x 9 /Эх and Э/Эх. One has [H ,K J = -K .

We say that a cochain a e C *( j^ ) is of weight p if L Ha = p a . Then

i) ak is of weight 1-k ( L HL K = - L K + L R L H);ii) if a is of weight p and J3 i s of weight q, then a A/3 i s of weight p + q;

iii) if a is of weight p, so a re da (L Hd = d L H ) and iHa ( iHL H = L H i ).

T h ere fo re C*(,jd) i s the d irec t sum of the hom ogeneous subcom plexes С Cs/), where C p(jd) i s the su b sp ace of coch ains of weight p.

H ow ever, fo r p ^0 ev ery cocycle in С (j¡/) i s a coboundary: if L Ha = pa and da = 0 , then p a = d iHa , so that the in clusion C 0(jrf)e-* C * ^ ) g iv e s r i s e to an iso m o rp h ism on the cohom ology grou ps.

The subcom plex C Q(j¡/) is z e ro in d egree la r g e r than 3, and is gen eratedby

1 in d egree 0 « j in d egree 1

a jA o j in d egree 2 agA ajA ttg in d egree 3

with

d ( « x ) = a0A a 2 d (l ) = d(aQA a 2 ) = d(aQA al A a 2) = 0

so that

Н кЫ ) = 0 fo r к =^0,3,

H V ) = R,

= R with the c la s s of aÂaÂa^ a s a gen era tor.

T h is show s (proposition 1) that the m orp h ism h * : H *(j ¡0 -► H *(M , R) depends only on ^

R E F E R E N C E S

[1] BERNSTEIN, I . , ROSENFELD, B .. F unc tiona l A nalysis (1972).[2] BOTT, R .. HAEFLIGER, A ., to ap p ear in B u ll .A m .M a th .S o c .[3] GELFAND. I . . FUKS, D ., C ohom ology o f the Lie a lg eb ra of fo rm al v ec to r fie lds. Izv . A k ad .N auk SSSR,

S e r .M at. 34 (1970) 322.[4] GODBILLON, C . , VEY, J . , Un in v a rian t des fe u ille tag es de cod im ension 1. С .R .A c a d .S c i.P a r is 273

(1971) A 92.[5] REEB, G . , Sur ce rta in e s p rop rié tés topo log iques des v a rié té s feu ille té e s , H erm ann (1952).[6] THURSTON, W ., N oncobordant fo lia tions o f S3 , B ull. A m .M ath .S o c . 78 (1972) 511.

/

IAEA-SMR-11/29

ON THE LOCAL SOLVABILITY OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS

H. GOLDSCHMIDTUniversité scientifique et m éd ica le de Grenoble, Grenoble, France

Abstract

ON THE LOCAL SOLVABILITY OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS.A survey o f lo c a ls o lv a b ili ty results for lin ea r inhom ogeneous p a r t ia l d iffe re n tia l equa tions w ith

С -co e ff ic ie n ts , as w e ll as cond itions for so lv ab ility and no n -so lv ab ility o f such equa tions , are given,

If x ,. . . . . x a re the co -o rd in a te s on H n, we se t1 * * n ’

and if a = (a , . . . , an) i s an n-tuple o f non-negative in te g e rs , we set

A d iffe ren tia l o p era to r o f o rd e r m on an open se t i 2 c E n i s a lin e a r o p era to r

where u i s a com plex-valued function o r d istribu tion and the a 0, a re com plexvalued C ~-functions on S7, and som e coefficien t a0, with |ar| = m does not van ish iden tically on Г2.

We sh a ll co n sid er the equation

w here f i s a given com plex-valu ed C“"-function on Q.

T h eorem 1 (C auchy-K ow alew ski). If the coeffic ien ts a0 of the o p era to r P of o rd e r m and f a r e an aly tic in a neighbourhood of x Qe f i and if the coeffic ien t of D™ is f 0 at x 0, th ere e x is t s an an aly tic solution u of E q . ( l ) on a neighbourhood of x 0.

T h eorem 2 (E h ren p re is - M algran ge). If the co e ffic ien ts a^ of the o p era to r P on ]Rn a re con stan t, then ,for a ll fG C ^ (Rn), th ere e x is t s a C“'- so lu tion u on lRn o f E q . ( l ) .

П

and

Pu = f (1)

221

222 GOLDSCHMIDT

The situation fo r C“ -co effic ien ts i s quite d ifferen t a s i s shown by the follow ing str ik in g exam ple of H ans Lewy (195 7) on IE3:

Pu = -iDjU + D 2u - 2(x1+ ix 2)D3u

w here i = . If f i s an an aly tic function on a neighbourhood of x 0eIR 3,acco rd in g to T heorem 1, th ere i s an an aly tic solution of E q . (1) on a neighbourhood o f x0. If f i s a function o f the v a r iab le x 3 alone, then Lewy proved by e lem en tary m ethods that, if f i s o f the form Pu on an open set U e IR 3 where u i s a C“ -function, f m ust be n e c e s sa r i ly analytic on U. C hoosing a С ““-function f o f the v a r ia b le x 3 which i s nowhere an aly tic , an o p era to r P and a function f a re obtained fo r which E q .( l ) h as no C ^-so lu tion and even no d istribution solution u on any non-em pty open su b se t of IR3.

With such a choice o f f, the hom ogeneous equation

Pu - fu = 0 (2)

1 3h as no n o n -triv ia l solution u e C (fi) on any non-em pty open su b se t fi of IR .If u ^ 0 on fi, then v = log u i s C 1 on som e open se t and s a t i s f ie s Pv = f,lead in g to a contradiction . If f w ere an aly tic , there would e x ist n on -triv iallo c a l an aly tic so lu tion s u o f E q .(2 ) accord in g to the Cauchy-K ow alew skiex isten ce theory.

R ecen tly , N iren b erg h as given an exam ple o f a f i r s t - o r d e r o p erato ro f the form

t 9 . , ,. 3L = a t +1 <p { x -

w here (x, t) a re co -o rd in a te s on IR2 and cp(x, t) i s a re a l-v a lu e d function on IR2, sa tis fy in g the following condition: any CT - solution u of the equation Lu = 0 on a connected open neighbourhood fi of the orig in in IR2 i s n e c e ssa r ily co n stan t.

L e t us in troduce an in varian t a sso c ia te d to P which p la y s a c ru c ia l ro le in the lo c a l ex isten ce theory of lin e a r d iffe ren tia l o p e ra to r s . The p rin c ipa l sym bol (or p rin c ipa l part) of P i s the polynom ial in Ç

P m(x, I ) = £ aa ( x ) ? °|n| =m

w here x e f i , f = ( f j ...........Ç j e E " and ? “ = f “ i . . . . i f a = K , . . . , <*„).We m ay co n sid er Pm to be a com plex-valued function on the cotangent bundle T * (fi) = fi X IRn of fi whose co -o rd in a te s a r e (x, f ) e f i X IRn. R e c a ll that i f cp, ф a re com plex-valued functions on T *(fi), we m ay define the P o isso n b rack et

, Л = V ( È<L M. ML ÈJL,v\ L \ S X j 3 f J ' 3 X j 3 ? 3

j = i

which i s a well-defined function on T * (fi).

IAEA-S MR-11/29 223

H orm ander h as given the follow ing n e c e s sa ry condition for the ex isten ce of so lu tion s:

T h eorem 3 . Suppose that E q .( l ) h as a d istribu tion solution u £ ® ' (П) fo r every fC C ” (Г2). Then

If f j = -2x2, S2 = ^x i ant 5з = 1. then P j (x, f ) = 0 and {P b P j } (x, f ) / 0; hence condition (3) does not hold fo r any non-em pty open su b se t £2 of IR3.

We now give the p r e c is e notion of lo c a l so lv ab ility .

D efinition . An o p era to r P i s sa id to be lo ca lly so lvab le in if fo r a ll x 0e fi, th ere i s an open neighbourhood UC£2of xq such that fo r any f e C " (U) th ere i s a d istribu tion u £ ® ' (U) solution o f E q . ( l ) .

We sh a ll co n sid er only o p e ra to rs of p rin c ipa l type.

D efinition. The o p era to r P of o rd e r m i s of p rin c ipa l type in Q if , fo r

and any zero of d^Pm w ill be a zero o f Pm of m ultip licity , at le a s t , two. In p a r t ic u la r , a ll e llip tic o p e ra to r s , i . e . th ose o p e ra to rs fo r which Pmv an ish es only if Ç = 0, and a l l hyperbolic o p e ra to rs a re of p rin c ipa l type; the p a r a bo lic ones a r e not.

Let us co n sid er the sim p le exam ple o f an o p era to r of p rin c ip a l type of the fo rm

w here (x, t) a re the co -o rd in a te s on IR2 and b(t) i s a re a l-v a lu e d C°°-function on som e open in te rv a l - T < t < T (T > 0). U sing F o u r ie r t ra n s fo rm s with re sp e c t to x , it is not d ifficu lt to show that P i s lo ca lly so lvab le in Г2 = { I x I < r , 11 1< T }, w here r > 0, i f and only if

If b(t) = tk, w here к i s a p ositive in teger, then P i s lo ca lly so lv ab le in Q i f and only i f к i s even.

We w ish to d e sc r ib e the condition o f N ire n b e rg -T rê v e s which extends condition (5) to a g e n era l o p e ra to r P o f p rin c ipa l type. We w rite

{P m< Pm[ (x - 5) = 0 w henever Pm (x, f ) = 0, (x, ? ) e T * (Q) (3)

In the c a se of L ew y 's exam ple

аП (x, f ) e T * ( Q ) , Ç f 0, one h a s dEPm (x, Ç) f 0.B y E u l e r 's identity , fo r hom ogeneous polynom ials

m Pm (x, f ) = Ç • g r a d E Pm (x, ?)

(4 )

b(t) does not change sign in the in te rv a l |t | < T (5)

Pm (x, f ) = A (x, f ) + iB (x, f )

where A, В a re re a l-v a lu e d functions on T * (Г2). Letn

ЭА а ЗА 3 3?: ЭХ: ЭХ: 3?, )

j= 1

224 GOLDSCHMIDT

be the H am iltonian v e c to r fie ld o f A on T * (Q). The in teg ra l cu rv e s of HA a re the so lu tion s of the equations

• _ Э_А • _ Э_АX ' Э?j ’ ' ‘ 3xj

and a re ca lled b ic h a ra c te r is t ic s t r ip s . Along such a cu rve, A is constant.If the value o f A on a s tr ip i s zero , then it i s ca lled a null b ic h a ra c te r is t ic s tr ip o f A .

The function HAB i s ju st the P o isso n b rack et {B , A }, so that H orm ander' condition (3) can be re fo rm u lated a s

(HAB ) (x, Ç) = 0 whenever Pm(x, f ) = 0, (x, f ) e T * (í¿)

o r a s :If В v an ish es at a point (x, Ç) o f a null b ic h a ra c te r is t ic s tr ip <if of A,

the f i r s t d eriv ative o f В a lo n g é van ish es at (x, §).L e t us return now to the o p era to r (4) or ra th er to - iP . Setting x = Xj,

t = x2, Ç = Çj, t = f 2, then A = т , B = b (t )Ç . The b ic h a ra c te r is t ic s tr ip s of A a r e p re c ise ly the stra igh t lin e s in iî X K 2 p a ra lle l to the t - a x is , and condition (5) can be re fo rm u lated a s :

В does not change sign on any null b ic h a ra c te r is t ic s tr ip of A.

We now state the N ire n b e rg -T rê v e s condition gen era liz in g the p rev iou s condition:

(N -T ) F o r a ll (x0, f 0)GT* (Q), Ç0 f 0, there i s a neighbourhood U of (xQ, Ç0) in T * (Г2) and a z £ С sa tisfy in g the following:

(i) d |R e (zP m) (x, Ç) / 0 for a ll (x, Ç )e U;(ii) the re str ic t io n of Im (zP m) to any null b ic h a ra c te r is t ic s tr ip of

Re (zPm) in U does not change sign .In verify in g th is condition for o p e ra to rs of p rin c ipa l type it i s enough

to take z = 1 or z = i . If P i s of p rin c ipa l type and Pm has r e a l coeffic ien ts, then th is condition holds a lw ays.

The follow ing re su lt w as obtained by N ire n b e rg -T rê v e s when the coeffic ien ts of Pm a re an aly tic in Q and by B e a ls - F e ffe rm a n in the general c a se :

T h eorem 4 . Let P be a d ifferen tia l o p era to r of p rin c ipa l type o f o rd er m in f2. If condition (N -T) holds, then P i s lo ca lly so lvab le in Q.

N ire n b e rg -T rê v e s a lso showed that if P m h as analytic co e ffic ien ts, then condition (N -T) i s a lso n e c e s sa ry :

T h eorem 5 (N ire n b e rg -T rê v e s ). Let P be a d iffe ren tia l o p era to r of p rin c ipa l type o f o rd e r m in Q. If Pm h as an alytic co e ffic ien ts, condition (N -T) holds if and only if P i s lo ca lly so lvab le in Г2.

R E F E R E N C E S

[ 1 ] HORMANDER, L. , Linear P artia l D iffe ren tia l O perators, S p ringer-V erlag , B erlin , G ottingen,H eide lberg (1963).

[ 2 ] TREVES, F. , On the ex istence and reg u la rity o f solutions o f linear p a r tia l d iffe ren tia l equa tions,Proc. Sym p. Pure M ath. , A m er. M ath. S o c . , P rov idence, R .I. (to appear).

IAEA-SMR-11/30

COMPACT OPERATORS AND THE MINIMAX PRINCIPLE

R. A. GOLDSTEIN, R. SAEKS Department of Mathematics,University of Notre Dame,Notre Dame, Ind.,United States of America

Abstract

COMPACT OPERATORS AND THE MINIMAX PRINCIPLE.In the present work, th e rudim ents o f a s tructu re theory for a class o f c o m p ac t operators on a

Banach space , based on a n o n -sp ec tra l m in im ax p rin c ip le , a re p resented .

1. INTRODUCTION

Since it s inception, the p re-em in en t problem of o p erato r theory h as been the form ulation of a stru c tu re theory fo r o p e ra to rs on an infinite - d im en sion al sp ace which extends, in a natural way, the c la s s ic a l stru c tu re th eory fo r m a tr ic e s . In the c a se of a com pact o p erato r on a H ilbert sp ace , such a gen era lizatio n is obtained by approxim atin g it with a sequence of o p e ra to rs of fin ite rank (i. e. fin ite-d im en sio n al ran ge), thereby allow ing one to deduce the s tru c tu ra l p ro p e rtie s of a com pact op erato r on a H ilbert sp ace from those of it s fin ite rank ap p ro x im ate s. U nfortunately, the lack of a su itab le m in im ax p rin cip le fo r the spectru m of a com pact o p era to r on a B anach sp ace , and hence an approxim ation theory, h as precluded the u se of such an approach in th is m ore gen era l setting . In the p re sen t work, the rudim ents of a s tru c tu re theory fo r a c la s s of com pact o p e ra to rs on a B an ach sp ac e , b a sed on a n o n -sp ec tra l th eo retic m in im iax p rin cip le a re p resen ted . H ere, the m inim ax p rin cip le is u sed to define a sequence of o p era to r w idths, which a fte r ap p ro p ria te n orm alization a re shown to p lay a ro le in the B anach sp ace theory s im ila r to that of the e igen valu es and s-n u m b ers of the H ilbert sp ace theory. In p a r t ic u la r , an approxim ation theorem and a tra c e c la s s concept fo r com pact o p e ra to rs i s obtained which n atu ra lly extends the c la s s ic a l H ilbert sp ace theory to Banach sp ace .

In the follow ing sectio n , the stru c tu re theory fo r com pact o p e ra to rs on a H ilbert sp ace i s review ed , f i r s t fo r positive h erm itian o p e ra to rs and then fo r gen era l com pact o p e ra to rs . T h ese r e su lts a re then em ployed in the follow ing sectio n to m otivate the concept of an op erato r width which a fte r n orm alization is u sed to form u late an o p era to r approxim ation theorem and to c h a ra c te r iz e v a r io u s id e a ls of tra c e c la s s o p e ra to rs . F in a lly , a num ber of open p ro b lem s a re d isc u sse d .

2. EIG EN V A LU ES IN H ILB E R T SPA CE

F o r an o p era to r , A , on a H ilbert sp ace , we define it s e igen valu es, in the u su a l m anner, a s the se t of com plex num bers, Xs. a(A), fo r which the o p era to r [A-A.] fa i ls to have a bounded in v erse . In g en era l, the com plexity

225

226 GOLDSTEIN and SAEKS

of the o p era to r A m a n ife sts it se lf in the com plexity of the se t o(A). In p a r t ic u la r ,

i) If A is bounded o(A) i s bounded.ii) If A is com pact c(A) i s countable with ze ro a s it s only accum ulation

point.iii) If A is se lf-ad jo in t ct(A) i s re a l.iv) If A is p o sitive cr(A) is p ositive .

A s such, the e igen valu es of a p o sitive se lf-ad jo in t com pact operator form a n o n -in creasin g sequence [2] of p o sitiv e r e a l num bers, A;(A)= 0, 1, 2, . . . , with the la r g e s t eigen value ( = ||a ||) taken a s \(A), the next l a r g e s t a s Xj(A), etc? Once the e igen valu es of A have been so ord ered , th e ir a lte rn ativ e c h arac te r iza tio n v ia the m in im ax prin cip le [2, 3] i s given by

X¡(A) = m inim um m axim um J|Аф - l | | ( 1 )LeLj IM -1

w here L¡ i s the se t of i-d im en sio n al su b sp ace s and ||А ф -ь|| denotes the d istan ce of the v ecto r Аф from the su b sp ace L . Now, with the aid of the above m inim ax th eorem , the stru c tu re of A follow s [2, 3]. In p a rticu la r ,

i) A h as the approxim ation p rop erty (i. e. A can be approxim ated inthe uniform o p era to r topology by a sequence of fin ite -ran k o p era to rs).

ii) A is a p - tra c e c la s s o p erato r if and only if X¡(A) i s an i p sequence.iii) A h as a d iagonal rep resen ta tio n

Ax = У Х .(А )< ^ ,х > ф . (2)

i = 0

w here фх i s the i-th e igen vector of A.Unlike the p o sitive se lf-ad jo in t op erato r con sid ered above, a gen era l

com pact o p era to r on a H ilbert sp ace need not have a p o sitiv e r e a l sp ectru m , hence the o rd erin g of e igen valu es which m ak e s the p reced in g developm ent p o ss ib le i s not ap p licab le in the gen era l c a s e . F o rtu n ate ly , how ever, a s tru c tu re theory fo r gen era l com pact o p e ra to rs can be deduced from the e igen valu es of it s "m agn itu d e". That is , one tak e s a p o la r decom position [1] of a given com pact o p erato r , A , a s

A = VM (3)

where M = -J A *A is p o sitive and h erm itian , and V is a p a r t ia l iso m etry , and u s e s the e igen valu es of M to ch arac te r iz e A. H ere, we denote by s ¡ (A) the i-th eigenvalue of M, i. e.

s¡ (A) = X¡(M) =X i (Æ *Â ) (4)

1 An e ig en v a lu e o f m u ltip l ic i ty m is rep ea ted m tim es in th e sequence ; h en c e , th e sequence is n on increasing but m ay not b e decreasing [ 2 ] .

IAEA-SMR-11/30 2 2 7

and term the p o sitive n o n -in creasin g sequence s¡(A ), i = 0, 1, 2, . . . , the s-n u m b e rs fo r A [2]. A s b e fo re , the b a s ic tool which allow s the s-n u m b ers to y ield a stru c tu re theory i s a m in im ax p rin cip le [2 ,3 ] yielding

s¡(A) = m inim um m axim um ||А ф -ь || (5)L<=L; М - 1

and, in turn, a s tru c tu re theory fo r the gen era l com pact o p era to r A. In p a rt icu la r :

i) A h as the approxim ation p roperty .ii) A i s a p - tra c e c la s s o p era to r if and only if s¡(A ) i s an i p sequence,

iii) A h as a "Sch m id t" rep resen ta tio n

Ax = У S i(A )<<^ i,x>0¡ ( 6)i= о

where Ф; i s the i-th e igen vector of M =\/A *A and =Vф1.

3. WIDTHS AND s-N U M B E R S IN BANACH SPACE

F o r com pact o p e ra to rs on a B anach sp ace the lack of an involution p re c lu d e s the p o ss ib ility of u sin g a p o lar decom position to reduce an a r b it r a r ily given com pact o p era to r to one with p o sitive e igen valu es a s w as done in H ilbert sp ace . F o rtu n ate ly , how ever, the rep re sen ta tio n of the s-n u m b e rs im plied by the m inim ax p rin cip le of E q . (5) d oes not em ploy the involution and is p e rfec tly w ell-defined fo r o p e ra to rs on a B anach sp ace . T h e re fo re , ra th er than attem ptin g to define the s-n u m b ers fo r a com pact o p e ra to r in Banach sp ace fro m its e igen valu es we u se the m inim ax p rin c ip le to define the s-n u m b e rs , though with an ap p ro p ria te n o rm alization [5]. F o r th is p u rp ose , we define the i-th width of a com pact o p erato r , A, on a B anach sp ace , X , by

dj(A) = infim um m axim um || Аф - b|| (7)L e4 M - i

and the i-th s-n u m ber by

s¡(A ) = Pi(X)d¡(A) (8)

where

Pi(X) = suprem um infim um ||e || (9)L e L ¡ E g El

where the infim um is taken ov er a l l p ro jec tio n s, E , on L . H ere, dj (A) i s ju s t the width of the im age , under A, of the unit b a ll of X , in the se n se of K olm ogorov [4] and the n orm aliz in g fa c to r , p ¡(X ), is determ in ed en tire ly by the B anach sp ace independently of A. F o r a H ilbert

228 GOLDSTEIN and SAEKS

sp a c e Pi (X ) = 1 fo r a ll i, hence our B anach sp ace definition fo r the s - n um bers co in cides with the c la s s ic a l definition of Eq. (4) [1]. In fac t,Pi (X ) = 1 i f and only if X i s a H ilbert sp ace [ 6] , hence the widths and s - n um bers a re indeed d ifferen t in Banach sp ace . In g en era l, the p¡ (X) s a t is fy , [ 5 ,7 ,8 ] ,

1 S p . (X) S i (10)

and m ay becom e unbounded. In fa c t , they grow lin ea r ly with i fo r L p andlp, P ^ 2 [ 8] .

Som e e lem en tary p ro p e r tie s of the widths and s-n u m b ers fo r com pact o p e ra to rs on a B anach sp ace a re a s follow s [4, 5]:

i) do(A) = s 0(A) = II a||.ii) di(A) i s a n o n -in creasin g sequence of p o sitive num bers which con

v e rg e s to ze ro if and only if A i s com pact.iii) d¡(A) = S j ( A ) fo r a ll i if and only if X i s a H ilbert sp ace .iv) pi(X) = Pi (X *) if X is re flex iv e .

Since in a B anach sp ace the w idths and s-n u m b ers d iffe r there a re two a lte rn a tiv e fo rm u latio n s of the tr a c e c la s s o p e ra to rs which gen era lize the H ilb e rt-sp a ce concept [2, 5]. H ence, we sa y that a com pact o p erator A on a Banach sp ace X is of c la s s Dp if d¡(A) i s an t sequence and we say it is of c la s s Sp if Sj (A) i s an sequence, 1 S p S ®. S im ila r ly , we denote by D„ the se t of o p e ra to rs fo r which d¡(A) i s a Co sequence and we denote by S^ the se t of o p e ra to rs fo r which S¡(A) i s a C 0 sequence. C learly ,

Sp С SqП П I S p S q S o o (11)Dpc D q

and by analogy with the H ilbert sp ace c a se [ l ] we m ay term an o p erator in c la s s D : d -n u c lear while c le a r ly the o p e ra to rs in D„ a re ju s t the com pact o p e ra to rs . S im ila r ly , the o p e ra to rs of c la s s S j m ay be term ed s-n u c le ar and those of c la s s ^ m ay be term ed s-co m p act.

4. AN APPROXIM ATION THEO RY

An o p e ra to r , A , on a B anach sp a c e , X , i s sa id to have the approxim ation p ro p erty if it can be approxim ated in the uniform o p erato r topology by a sequence of o p e ra to rs of fin ite rank, and the B anach sp ac e , X , is sa id to have the approxim ation p ro p erty if ev ery com pact o p erato r on X h as the approxim ation p roperty . Although, (a s once con jectured by Banach), it i s not true that ev ery com pact o p era to r h a s the approxim ation property [9], it is p o ss ib le to show that the s-co m p act o p e ra to rs do indeed have the approxim ation p ro p erty [5].

T h eo rem : E v e ry s-co m p ac t o p era to r h as the approxim ation property . P ro o f: L e t L¡ be a su b sp ace of X such that

+ dj (А) й m axim um ||Аф - l || (12)

GOLDSTEIN and SAEKS 229

L et L | be it s ann ih ilai or in X * , and let E¡ be a p ro jectio n onto L ; sa tis fy in g

| м . { Г Д и т и } + 1 1131i

Now E^ = 1 - E r i s a p ro jectio n with ran ge in ii. fo r if f = E ^ g =(1 - E r ) g and x is in L j then

< f ,x > =<(1 - E f ) g ,x > = < g ,( l - E. )x> =<g, x -x > = 0 (14)

sin ce E¡ is a p ro jectio n onto L ¡. M oreover,

I K II = 111 -E ? IIS 1 +||E?II = 1 + IIE¡ IIS 2 IIeJI S 2 ( j f e™ um | |f | |+ l } *2 f a (X) = l }

(15)

w here the la s t inequality r e su lt s from the fac t that the suprem um in the defin ition of p¡(X) i s taken o v er a l l i-d im en sio n al su b sp a c e s of X includingL ¡ . Now upon recon gizin g [10, 11] that the d istan ce between the vecto r А фand the su b sp ace Ц is given by

I N - L j I = m a x im u m |< f , A ^>| . . . .f e L t ( 1 6 )

N i

and that the m inim um in E q , (12) i s a lm o st ach ieved by L if we have

+ di m axim um II Аф - l J = m axim um m axim um |<f, A<¿> I 1 1 И И ! !® N l feL j-

Ilf II ^ 1 1 '

= m axim um m axim um |< A *f, ф > I = m axim um ||A *f || f eLj H0II-1 feL¡

Ilf IN I ||f II s i

H ere, the f i r s t equality i s that of Eq . (12) with the m inim um achieved fo r the sp ec ified L j , the second re su lt s from substitu tion of the d istance fo rm u la of Eq. (15), the third r e su lt s from in terchanging the o rd er of the m ax im a and the definition of the ad joint, and the la s t equality i s ju s t the definition of the norm [1] of the functional A *f . Now, sin ce the ran ge of Ej1 i s contained in L ¡ and the o p era to r Е ^/Ц е1!! h as norm one we obtain the se t containm ent

{ f e X : f = ( E fg ) / | |E , i , | | g | | s i } c { f e X : f e L i , M s 1} (18)

230 IAEA-SMR-11/ЗО

and hence Eq. (16) becom es

-i- + d j(A )S m axim um ||A *f || й m axim um { ||A>:'E^g|¡ / Це ^Ц} 1 llgll s i

= [ || A *E f ||/ ||e +|| ] = [ IIA*( 1 - E f )II /¡E fll ] = [ II( 1 -E, )а ||/||е ^||] (19)

= [|| A -E j a || / ||Ef|| ]

F in a lly , upon com bining E q s (15) and (19), we have

A -E í AH s | |E iJ'||(di ( A ) + I |- ) S 2{Pj (X)+ 1} (d,(A) + T|-)

(20)

. 2 , 2Pi (X) i2 i2 + 2 s. (A) + 2d. (A) S 4s. (A) + 2 (1 1 - + —:

•)

Hence if A is s-co m p ac t the s¡ (A) converge to zero im plying that the sequence of fin ite-ran k o p e ra to rs E¡ A con v erges in the uniform o p erator topology to A , com pleting the proof of the theorem .

The theorem h as a num ber of im m ediate c o r o lla r ie s where under ap p ro p ria te assu m p tio n s a c la s s of o p e ra to rs m ay be shown to be s - com pact and thus have the approxim ation p roperty . In p a r ticu la r , sin ce Sp£ S „w e im m ediate ly obtain:

C o ro lla ry : E v e ry o p erato r of c la s s Sp, U p S a h as the approxim ation property .

Although the Dp c l a s s e s a re not, in gen era l, s-co m p ac t it i s p o ssib le to show that the o p e ra to rs of c la s s Dj a re s-co m p ac t fo r if dj(A) i s an sequence then id¡(A) go es to ze ro and sin ce p ¡(Z ) § i we have

which con v erges to zero a s i g o e s to infinity. Hence s¡(A ) i s a Q sequence if d¡(A) is an sequence and we have:

C o ro lla ry : E v e ry d -n u clear o p era to r h as the approxim ation p roperty . F in a lly , if the p¡(X) sequence is bounded the every com pact op erato r is s-co m p ac t sin ce the d¡(A) go to ze ro fo r com pact operators [4, 5] and we have:

C o ro llary : E v e ry B anach sp ace X with a bounded p¡(X) sequence h as the approxim ation property .

C le a r ly , the c o ro lla ry y ie ld s the c la s s ic a l re su lt that ev ery H ilbert sp ace h as the approxim ation p rop erty [2].

s¡ (A) = Pi(A )di(A M id.(A ) (21 )

GOLDSTEIN and SAEKS 231

A s with th e ir H ilbert sp ace an tecedents [2], the tra c e c la s s o p e ra to rs Sp and Dp form id e a ls in the a lg e b ra of bounded o p e ra to rs on a Banach sp ace . F o r th is pu rpose we equip Dp with the norm

5. O P E R A T O R I D E A L S

1 /p1 S p < oo

and Sp with the normi = о

(22 )

Г .°°

Liip = Y , Si(A)Pi/p

1 S p S oo

1 = 0

while Dœ and S a re norm ed by

IIAjj = m axim um {d¡(A )} = d0(A) = ||A|| °° о si -= «>

and

(23)

(24)

IA IL = m axim um { s ; (A)} Os i

(25)

re sp e c tiv e ly . We then have:

T heorem : The tra c e c la s s e s S p and Dp, I S p S » of o p e ra to rs on aB anach sp ace X a re norm ed by || ||p and || ||p, re sp e c tiv e ly , and a re c lo sed tw o-sided id e a ls in the a lg e b ra of a l l bounded lin e a r o p e ra to rs on X. The theorem re su lt s from stan dard techniques, analogous to those u sed on H ilbert sp ace [2, 3], and it s p roof w ill th erefo re not be given h ere. S im ila r ly , techniques analogous to those u sed in H ilbert sp ace yield ch arac te r iz a tio n s of the d u als of the Sp and Dp c la s s e s and m ost of the other p ro p e r tie s which one a s s o c ia te s with tra c e c la s s o p e ra to rs [2 ,3 ] . A s such , the Sp and Dj, c la s s e s a r e , indeed, natural g en era liza tio n s to Banach sp ace of the c la s s ic a l tra c e c la s s o p e ra to rs .

6 . CONCLUSIONS

Our pu rpose in the p reced in g h as been to indicate one m ethod by which the stru c tu re theory fo r com pact o p e ra to rs on a H ilbert sp ace can be extended to B anach sp ace . The key to the m ethod i s the u se of the m inim ax p rin c ip le , ra th e r than e igen valu es, a s the e lem en tary concept. T h is y ie ld s a v iab le g en era lizatio n , though one which is not sp e c tra l th eoretic in nature, which n atu rally extends m uch of the H ilbert sp ace theory to Banach sp ace .In p a r t ic u la r , it h as been shown that:

i) an s-co m p ac t o p era to r . A , h as the approxim ation p roperty ; ii) the p - tr a c e c la s s o p e ra to rs form c lo sed , tw o-sided id e a ls in the

a lg e b ra of bounded o p e ra to rs .

232 IAEA-SMR-11/3 О

Unlike the above r e su lt s , a s a t is fa c to r y gen era lizatio n to Banach sp ace of the Schm idt rep re sen ta tio n h as yet to be found though we con jecture that such a rep re sen ta tio n e x is t s fo r s-co m p ac t o p e ra to rs in the form

where the a re , a fte r ap p rop ria te norm alization , the ex trem al v e c to rs ф. which m ax im ize ||Аф. - L.|| and 0. = Аф.. S im ila r ly , there a re a num ber of open p ro b lem s a s so c ia te d with the approxim ation p ro p erty . In p a rt icu la r , s -c o m p a c tn e ss i s a n e c e s sa ry condition fo r an o p erato r on a B an ach sp ace to have the approxim ation p rop erty and the condition that p¿(X) be bounded is n e c e s sa r y fo r X to have the approxim ation p roperty .

[1 ] RIESZ, F. , S Z .-N A G Y , B. , F unctional A nalysis, New York, U n g ar(1 9 5 5 ).[ 2 ] GOHBERG, I . C . , KREIN, M .G . , In troduction to th e T heory o f L inear N on se lf-ad jo in t O perators,

P rovidence, Am . M ath . Soc. (1969).[3 ] DUNFORD, N. , SCHWARTZ, J. , L inear O perators, 2, New York, W iley (1963).[4 ] LORENTZ, G .G . , A pprox im ation o f Functions, New York, H olt, R inehart and W inston (1966).[ 5 ] GOLDSTEIN, R. A ., SAEKS, R. , "T ra ce class, w idths and the f in ite app rox im ation p roperty” , Bull.

Am . M ath. Soc. (to appear).[6 ] KAKUTANI, S. , "Som e C h ara c te riza tio n s o f Euclidean S p ace” , Proc. M ath . Soc. Japan 16 ( 1939) 93.[7 ] GOLDBERG, S . , U nbounded L inear O perators, New York, M cG raw -H ill (1966).[ 8 ] MURRAY, F . J . , "On co m p lem en ta ry m anifo lds and pro jections in spaces Lp and Í T rans. Am.

M ath . Soc. 40 (1937).[9 ] ENFLO, P . , unpublished notes.

[1 0 ] HOLMES, R. B ., A Cours in O p tim iza tio n and Best A pprox im ation , New Y ork, S pringer-V erlag (1972).[1 1 ] LUENBERGER, D .G . , O p tim iza tio n by V ector S pace M ethods, New York, W iley (1969).

(26)

i = 0

R E F E R E N C E S

IA EA-SMR-11/31

RIGIDITY AND ENERGY

R .A . GOLDSTEIN, P.J. RYAN Department o f M athem atics,U niversity o f Notre D am e,Notre D am e, Ind. ,U nited States o f Am erica

Abstract

RIGIDITY AND ENERGY.V arious concep ts o f r ig id ity are p resen ted and il lu s tra ted . Am ong o ther ite m s , in fin ite s im a l r ig id ity is

fo rm u la ted g lo b a lly and th e d iffe re n tia l eq u a tio n for an in f in ite s im a l iso m e tric defo rm ation is g iven .

1. RIGIDITY

If you p icture a R iem annian m anifold, you a re see in g it im m e rse d in som e E u clid ean sp a c e . Two q uestion s n aturally a r i s e . F i r s t , given a R iem annian m anifold M, can it be iso m e tr ic a lly im m e rse d in som e p a rt ic u la r E u clidean sp a c e ? T h is i s a question of ex isten ce o f a m ap with sp e c ific p ro p e r tie s . In th is p ap er, we a re in te re ste d in the other question— uniqueness of th is m ap when it e x is t s . In other w ords, when you p icture a ce rta in R iem annian m anifold M, i s what you se e the only p o ss ib le p ic tu re , say , up to an iso m etry o f the E u clidean sp a c e ?

Definition: A R iem annian m anifold M which can be iso m e tr ic a lly im m e rse d in E n i s sa id to be r ig id in E n if for any two iso m e tr ic im m e rs io n s г : and r2, th ere i s an iso m etry <p o f E n such that r 2 = <p о r j .

We l is t som e well-known r e su lts concerning rig id ity :i) The stan d ard sp h ere S 2 i s r ig id in E 3. In fac t, every c lo sed su r fa c e

in E 3 with G a u ssian cu rvatu re К ê 0 is r ig id .ii) Not much is known about su r fa c e s w hose curvatu re changes sign .iii) When n s 4, every h y p ersu rface with positive sec tio n a l curvature

i s r ig id . T h is i s a consequence o f the rank of the second fundam ental form being è 3 and i s a much e a s ie r re su lt than i).

One might con jecture that every c lo se d su r fa c e in E 3 i s r ig id . T h is is f a ls e a s shown by a cou n ter-exam ple o f C oh n -V ossen . F o r in stan ce , if a c lo se d su rfac e has a p lan ar p iece , one can con struct a C " bum p lying on one sid e o f the p lane. Since th is bump i s iso m e tr ic to its m ir ro r im age in the p lane, one can con struct two iso m e tr ic but non-congruent su r fa c e s . F o r an illu stra tio n , see Stoker [7 ] p . 366. S tok er co n jectu res that any non- r ig id c lo sed su r fa c e m ust have th is re flec tio n p roperty at one o r m ore bum ps.

We a re a ssu m in g that a ll m anifo lds a r e connected. The sym bol x (M) w ill denote the L ie a lg e b ra of a ll v ecto r f ie ld s on M and I(M) i s the iso m e try group of M. We a ssu m e a l l su r fa c e s and m aps a re su ffic ien tly d ifferen tiab le , say C 4. It i s a fact [ 5] that the sph ere S2 i s not C 1-r ig id but i s C 2-r ig id .

233

234 GOLDSTEIN and RYAN

In th is sec tio n , we d isc u s s a second type of rig id ity m ore consisten t with the d iction ary m eaning o f the w ord, i . e . r e s is ta n c e to bending. We ca ll th is concept continuous r ig id ity .

D efinition: Let S = (M, r ) be a subm anifold o f a R iem annian m anifold M.Thus r : M -* M is an im m ersio n . Let I = [-1, 1] . A m ap

7 : I X M - M

i s a deform ation o f S if 7 0 = r and 7t is an im m ersio n for each t . H ere 7t (x) = 7 (t, x).

E ach im m ersio n 7 t in duces a R iem annian m etric g t on M and each c lo sed curve on M h as a length L(t) m easu red by the m etric g.

D efinition: Let 7 be a deform ation o f S . Then 7 i s an iso m e tr ic deform ation (ID) o f S i f gt = g 0 for each t.

Definition: Let S = (M, r ) be a subm anifold o f a R iem annian m anifold M.Then S is^ sa id to be continuously r ig id if for every ID of S th ere i s a curve <p{t) in I(M ) such that fo r a l l t and x

7 t (x) = <p(t) r (x )

C le ar ly , S i s continuously r ig id if M i s r ig id . On the other hand,C o h n -V o ssen 's exam ple is continuously r ig id but not r ig id .

The follow ing con jectu re i s s t i ll open: E v ery c lo se d su r fa c e in E 3 i s continuously r ig id .

2 . C O N T IN U O U S R I G I D I T Y

3 . IN FIN ITESIM A L RIGIDITY

We now com e to the lin ea r iz ed v e rs io n of rig id ity (a notion used in e la s t ic ity theory) ca lled in fin ite sim al r ig id ity . Roughly speak in g, the in fin ite sim al th eory i s ju st the continuous theory with te rm s of second and higher o rd e r n eg lected .

F o r exam ple , if S i s a su rfac e in E 3 the analogue of an ID i s an in fin ite sim al iso m e tr ic deform ation (IID) which i s a deform ation 7 with line e lem ents sa tisfy in g

(dYt )2 = (d T0)2 + O (t2)

M ore sp e c ific a lly , in the notation o f the la s t sectio n , we have

D efinition: A deform ation 7 i s an HD if g '(0 ) = 0. H ere , we a r e regard in g gt a s a cu rve in the fin ite-d im en sio n al v ec to r sp ace o f (0, 2) te n so rs at a point o f M. Note that th is a lso m ean s that L '(0 ) = 0 fo r each c lo se d curve on M.

D efinition: A subm anifold S = (M, r) of E n is in fin ite sim allÿ r ig id (IR) if fo r every IID 7 , th ere e x is t s a curve <p (t) e I (E n) such that

7 t (x) = <p(t) r (x ) + 0 ( 0 (1)

IAEA-SMR-11/31 235

F o r any deform ation y, we can w rite

If Yt - cp(t) о r , then it i s e a sy to v erify that

z = a r + b (3)

for som e constant sk ew -sy m m etric m atr ix a and constant v ec to r b . One can a lso v erify that a subm anifold i s IR i f and only if the v ec to r z determ ined by (2) i s o f the form (3).

The b e st known re su lt on in fin ite sim al r ig id ity is the follow ing theorem due to Liebm ann.

T h eorem : E v e ry c lo sed su rface in E 3 with K ê О (К not iden tically zero on any open se t) i s in fin ite sim ally r ig id . In p a r t ic u la r , i f К > 0 the conclusion h o ld s.

We se e that the hypoth esis К i 0 i s not su fficien t fo r in fin ite sim al rig id ity o f c lo se d su r fa c e s by a m odification of the next exam ple .

E x am p le . Let S = (M, r ) be the open unit d isk in E 2 with r the in clusion m ap into E 3. Let

7 t (x ,y ) = (x, y, t ( l - x 2-y 2)) ( t e [ - 1 , 1 ] )

Then у i s an IID but not an ID. F u rth e rm o re , the correspon din g z cannot com e from a E u clid ean m otion. Thus S i s not in fin ite sim ally r ig id .

4 . G LO BA L FORM ULATION OF IN FIN ITESIM A L RIGIDITY

L et S = (M, r) be a subm anifold of M. L e t E be the re s tr ic t io n to M of the tangent bundle T (M ). The se t o f sec tio n s of E i s denoted by Г (Е ) and such a sectio n i s c a lled a v ecto r fie ld along r . E b eco m es a R iem annian vec to r bundle by re s tr ic t in g the m etr ic of M to the fib re s o f E . If V i s the connection on M, one defines a connection D on E by

(Dx u) (p) = (Vx u) (r(p))

w here X e x(M ), и £ Г (Е ) , p e M and X and u a re v ec to r f ie ld s extending r „ X and u, re sp e c tiv e ly , on som e neighbourhood o f r(p ).

F o r и е Г (Е ) , the ex te r io r deriv ative du of u i s the E -v a lu e d 1-fo rm defined by

(du) (X) = Dxu

A lso , i f в i s an E -v a lu ed 1 -fo rm , then E -v a lu ed 2 -fo rm dg i s defined by

d в (X , Y) = Dx (6 Y) - Dy (0X ) - 0 [ X , Y]

Tt = 7 o + t z + 0 ( t 2) (2)

We note the re lationship

d (du) (X, Y) = R (X, Y) u

2 3 6 GOLDSTEIN and RYAN

where R i s the cu rvatu re te n so r of M. F u rth erm o re , if M i s E uclidean , we have

d2u = 0

fo r a ll u e r ( E ) .

D efinition: If 7 is a deform ation of S , then the deform ation v ecto r field z i s the sectio n of E w hose value at x i s the in itia l tangent v ecto r to the curve

t - 7 t (x)

Thus zx i s the " in it ia l v e lo c ity " o f x under the deform ation .

T h eo rem : 7 is an IID if and only if for a ll X , Y in *(M ),

< D x z, Y > + < X , Dy z > = 0 (4)

The E -v a lu ed 1 -fo rm p = dz i s ca lled the rotation fo rm . Note that 7 isan IID if and only if

< pX, Y > + < X , pY > = 0

We r e s ta te the definition o f in fin ite sim al rig id ity sin ce we a re now allow ing M to be non-E uclidean .

D efinition: S i s IR i f fo r each zG F (E ) sa tis fy in g (4), th ere i s a cúrve i/>(t)eI(M ) such that the m ap

(t,x ) - <p(t) r (x)

i s a deform ation o f S with deform ation v ecto r z. In such a c a se , we say that z i s t r iv ia l .

R e m ark : When M is E u clid ean , z i s t r iv ia l i f and only if it i s of the form (3).

5 . E X A M P L E S AND CO U N TER EX A M PLES

In th is sec tio n , we sta te se v e ra l th eo rem s concerning r ig id ity o f h y p ersu r fa c e s o f h igher d im ension . P ro o fs m ay be found in R ef. [ 3] .

T h eo rem : The h yp ersp h ere S n(R) o f ra d iu s R in E n+1 i s IR .

T h eo rem : If R Q < R, S n(RQ) in S n+1 (R) i s IR.

T h eo rem : Any h y p ersu rface o f E n + 1 som e open se t of which l ie s in a h yp erplane adm its a n o n -triv ia l IID.

T h eo rem : The g rea t sph ere S n(R) in S n + 1 (R) i s not IR .T h is la s t re su lt c o n tra sts with the th eorem of O 'N e ill-S tie l [ 6 ] which

s a y s that S n(R) in S n + 1 (R) i s r ig id in the se n se o f section 1.

lAEA-SMR-11/31 237

L et S = (M, r) be a subm anifold o f M. Suppose M and M have d im ensions n and n + p , re sp e c tiv e ly . An orthonorm al se t {et} " Í Ç of n + p sec tio n s of E i s c a lled an adapted fram e on S i f the f i r s t n sectio n s a re tangent to M.

P ro p o sitio n . Let {e ;}"^Ç be an adapted fram e and let be the co fram eon M dual to |e j}"=1. Then a s a sectio n of E ,

6. THE D IF F E R E N T IA L EQUATIONS FO R AN IID

R e m a rk : To explain notation, if s e r ( E ) and a i s a 1 -fo rm on M, then a s i s the E -v a lu ed 1-form defined by

(as) (X) = a (X )s (X e *(M ))

F ro m now on, we take n = 2 and M = E 3. The ord in ary c r o s s product in IR3 defines a m ap

Г (Е ) X Г (Е ) - Г (Е )

which extends to a l l E -v a lu ed fo rm s.Now su ppose z i s a deform ation v ec to r fie ld sa tis fy in g (4). The rotation

fo rm p = dz g iv es r i s e to a sectio n Р е Г ( Е ) ca lled the rotation fie ld such that fo r X e x (M),

p X = P X X

L em m a: dP h as no n o rm al com ponent.

C o ro lla ry : T h ere ex ist 1- fo rm s T j and t 2 on M such that

dP = г1е1 + т2 е2

and

(5)

Tia2 - T2CTi = 0

H ere juxtaposition of fo rm s stan d s fo r the ord in ary wedge product.\

C o ro lla ry : T h ere ex ist functions a , J3 and у on M such that

(6 )

T1 _ a & alK - .У -a l°-2j

Let ш be the connection form and let

0-ш


Let cjj and u 2 be fo rm s (e sse n tia lly , the second fundam ental form ) defined by the equation

de = Qe - u e3

w here e = e i and u = U)iLe 2- -U2-

R eca llin g (5), we se e that d2P = 0 im p lie s

dr - ftr = 0 (7)

andt * u = 0 ( 8 )

It is often p o ss ib le to choose a fram e so that the v e c to r s a re principal and E q s (7) and (8) red u ce to a s im p le r fo rm . Such i s the c a se in the follow ing exam ple:

S u rfa c e s of R evolution: C on sider the su r fa c e o f revolution o f the graph of a function g p a ram e tr ize d by r and 0 a,s fo llow s:

x = r c o se у = r s i n 0 z = g(r)

A g lobal p rin c ipa l adapted fram e is given by

e x = (- sin 0, cos 0, 0)

e 2 = £ (cos 0, s in 0, p)

e 3 = j ? ( p c o s 0, p s i n 0 , - 1 )

w here

Г 1 = n/ 1 + p2 and p = g' ( r )

(jj = rd 0 dcj = drd 0

erg = i _1dr dcr2 = 0

и = -Hr _1 Oj

Wl = рш 102 = - P 1 3k j = - p i r "1 k2 = - p ' i3

к = -E-f (1 + p2)rp '

H ere, k 2 and k 2 a re p rin c ipa l c u rv a tu re s . We denote th e ir ra tio k x/ k 2 by k. W henever { e x, e2} i s a p rin c ipa l fram e , we have

Uj = kjffj and u)2 = k2a 2

kj/3 - k 2 Y = °

In view of (8), we have the identity

T h us, provided that к i s w ell-defined , we can w rite (7) in the form

-T df + (dT - fiT ) f = 0

where

IAEA-SMR-11/31 239

f = a and T = °1 C2A - ' CT2 kcjj

M ultiplying by 1 0 0 -1

and settin g

A = °1 CT2 -ka

we have

which we ab b rev iate

-A df + (dA + ÍJA) f = 0 (9)

L f = 0

Now L f i s a 2 X 2 m atr ix o f 2 - fo r m s . The in fin ite s im a l-r ig id ity problem now b o ils down to the follow ing question :

D oes the d ifferen tia l o p era to r

L = -Ad + (dA + fiA)

adm it non -zero so lu tion s fo r L f = 0?

7. IN FIN ITESIM A L RIGIDITY WITH BOUNDARY CONDITIONSО

N on-com pact su r fa c e s in E a r e se ldo m IR a s such. It i s n atu ral in a ph ysical situation to co n sid er holding certa in p a rts of a su r fa c e fixed while bending the rem ain d er.

T h u s, we sh a ll allow our su r fa c e s to be oriented tw o-d im ensional m anifo lds with b ou n daries contained in la r g e r s u r fa c e s . A ll deform ation s a re a ssu m ed to be defined on the la r g e r m an ifo ld s.

In th is situation , an IID d eterm in es and i s determ in ed by a fie ld z sa tis fy in g (4).

D efinition: Let S = (M, r) be a su r fac e in the sen se of the above p arag rap h . If M 0C M, then S i s IR mod M 0if d z |M 0 = 0 im p lie s z is t r iv ia l .

R e m ark : In our ap p licatio n s, M0 i s a portion o f the boundary o f M.

D efinition: In te r m s of the o p e ra to rs in troduced in sectio n 6 , we define the deform ation energy o f M to be the m a tr ix of 2 -fo rm s

к - dA + fiA + A fi


In c a se к is not defined everyw here by th is fo rm u la , we se t к = 0 on the rem ain d e r . Then we have the identity

2 < L f . f > M = <Kf, f > M+ < A f, f > 9M (10)

w here the "in n er p rodu ct" <(0, ip ^Mis defined by in tegrating over M the 2 -fo rm

01^ 1 + Э2Ф 2

obtained fro m the m atr ix of 2 fo rm s

"e ll and the m a tr ix of functions Ф 1- 02- _Ф2-

The e x p re ss io n <( A f, O g Mi s evaluated s im ila r ly .

D efinition: к i s sa id to be p o sitive on M if <K f, О м i s p o sitive fo r a l l nonzero functions f. N egativ ity o f к i s defined an alogously .

R e m ark : If two d ifferen t d eterm in ation s of к a r i s e from two d ifferent p rin c ipa l fr a m e s , the p rop erty of d e fin iten ess (positiv ity o r negativity) i s p re se rv e d .

We a lso define p o sitiv ity of A pointw ise on Э M. Suppose j : S 1 -» M is a cy c le in ЭМ. The (j'"A) (Э/ 9 t) i s a m atr ix of functions on S 1. We choose t to a g re e with the orien tation of ЭМ induced by that o f M. Then A is p o sitiv e at j(t) i f the a s so c ia te d m atr ix i s p o sitive definite at t . The portion o f the boundary where A is not positive is w ritten 9M_. The se t 9M+ i s defined an alogously .

One can deduce the following theorem from (10):

T h eo rem : Let S be a com pact orien tab le su r fa c e (with boundary) in E 3. If к i s defined and definite then S i s in fin itesim ally r ig id mod 9M . o r 9M + .

T he energy m ethod u sed h ere i s s im ila r to that of F r ie d r ic h s [ 2 ] . We have a p o sitiv e sy m m etric sy ste m of d iffe ren tia l equation s. H ow ever, in our c a se , the co e ffic ien ts a re fo rm s ra th e r than functions. We se e that к m ay be definite even though the G a u ssian curvatu re changes sign . Ju s t as F r ie d r ic h s can tre a t equations which a r e e llip tic in one reg ion and h yperb o lic in another, we can trea t su r fa c e s whose cu rvatu re i s positive in one reg ion and n egative in another.

We note that the sym bol of the o p era to r L i s equal to -A d. T hus L is e llip tic (and p arab o lic and hyperbolic, re sp e c tiv e ly ) i f and only if the G au ssian cu rvatu re i s p o sitive (and zero and n egative, re sp e c tiv e ly ) .

E x a m p le : The caten o id . The curve у = co sh "xx i s revolved around the у a x is and S i s the su rfac e of revo lution . In the notation p rev io u sly used for su r fa c e s of revolution , we have

1p = ———■J r 2 - 1

k = -1

IAEA-SMR-11/31 241

(Note that H = k x + k 2 = m ean cu rvatu re = 0, o r the catenoid is a "m in im al s u r f a c e . ")

dA = I dr d0

fi =ч/rZ- 1 0

-cr-,

A fi = fiA = r 2 - 10 -1

2s/ r 2 - 1

a l CT2

1 -2ч/ r 2 - 1

The determ inan t of th is m atr ix i s

r.-4 (r2 - 2)2

which show s that к i s p o sitiv e .If we con sider portion s of the su r fa c e bounded by two cu rv e s of the

fo rm r = const, then

(j A ) (90) -

B e c a u se o f orien tation , A w ill be p ositive on one boundary curve and negative on the o th er. T hus, i f one boundary curve i s held fixed during an IID, the corresp on d in g solution to L f = 0 m ust be t r iv ia l .

One can so m etim es re p la ce a given problem fo r which к i s not positive by an equivalent prob lem fo r which к i s p o s it iv e . Let a , b and X be re a l functions on M such that a 2 + kb2 f 0. Let

V = -kba

Then we can define a new o p era to r L y x by

L v x g = L (e Vg) ( И )

and a r r iv e at a new en ergy te rm kv x sa tis fy in g an equation analogous to E q .(1 0 ) . Then if

g

x

Г 2 V _1f

p ositiv ity of kVi togeth er with a boundary condition w ill im ply g = 0 and hence f = 0. The following is an exam ple of th is technique:


E x am p le : the p arab o lo id . The curve y = x2 i s revo lved around the y a x is and S i s the re su ltin g su r fa c e of revo lution . U sing the fram ew ork of sectio n 6 , we have

A == CT1 n = —'0 -CTl

a2 -ktjj r .CTi 0

-f i -! Oj 02 = £2 A

dA = - r1 0 0 8r 2 J1

IК = —r

-1 00 8 ^ 2 CT1 °2

-2(1 + 4 r 2)1/2

k2 = '-2

(1 + 4Г2 )3 /2

к = 1 + 4Г2

Thus к i s not d efin ite . But now setx

g = e 2 f

F o r th is exam ple , we sh a ll be using X = - (3/2) ln r , but work with a gen era l X fo r a while in o rd er to b etter il lu s tr a te the m ethod. Note that we have taken V = I in (11). In th is c a se we su p p re ss m ention of V in our notation. M ore sp e c ific a lly , we have

A h L f = L (e2 g) = e 2 (Lg - i A dX g)

W riting

L x g = Lg - I A d X g

and

= к - Ad X

we deduce with a com putation that

2 < L x g, g > M= <K x g, g \ + < A g , g > aM (12)

If it i s known that L ^ g = 0, kx i s p o sitive and g = 0 on ЭМ, then g m ust be id en tically ze ro .

F o r the p arab olo id

L ^g = e "x/2 L f = 0

IAEA-SMR-11/31 243

A lso ,

-AdX = 3Í2r

<*i-ff2

°2-kcr, dr

Thus

-AdX =

Kx =

M2r

2r

1 0 0 - (1 + 4 r2) °1 <2

01 + 4 r2 CTj 02

which i s p o sitiv e . T hus we have the follow ing theorem :

T h eo rem : The portion M of the p arab olo id y = x 2 + z2 lying betw een two p lan es y = constant i s IR mod 3M.

R e m ark : By furth er m odification o f L^ , one can fo rc e the boundary in tegra l in (12) to be positive on one of the boundary c u rv e s . T hus only one of the boundary conditions i s n e c e s sa ry .

The following th eo rem s can be proved u sin g techniques s im ila r to those fo r the catenoid and parab olo id :

T h eo rem : E v e ry m in im al su r fa c e w hose flat points (if any) a re iso la te d i s IR mod ЭМ.

T h eo rem : Let g be a function whose graph y = g(x) has only fin itely many points of in flection . Then the su r fa c e o f revolution of g about the y a x is i s IR m od 3M.

T h eo rem : Let S be a developable su r fa c e whose fla t points form a se t ofm e a su re ze ro . Then S i s IR m od ^ where i s any c lo sed curve tr a n sv e r sa l to the g e n e ra to rs of S .

R E F E R E N C E S

[ 1 ] EFIMOV, N .V . , Q u a lita tiv e p roblem s in th e theory o f defo rm ation o f surfaces, D iffe ren tia l g eo m etryand ca lcu lu s o f varia tio n s, AMS tran sla tio n s, S er. 1, £ , 274.

[ 2 ] FRIEDRICHS, K. O . , S ym m etric positive lin ear d iffe re n tia l equa tions . C om m . Pure A ppl. M ath. 11(1958) 333.

[ 3 ] GOLDSTEIN, R .A ., RYAN, P .I . , In fin ite s im a l r ig id ity o f subm anifo lds (to appear).[ 4 ] GOLDSTEIN, R. A . , RYAN, P . J. , In fin ite s im a l r ig id ity and th e energy m ethod (to appear).[ 5 ] KUIPER, N. , On C 1 isom e tric em bedd ings, Proc. M ath. A cad. S c i. S er. A 58 (1955) 545, 683.[6 ] O'NEILL, B., STIEL, E. , Isom etric im m ersions o f constan t cu rvatu re m anifo lds, M ich. M ath. J. 10

(1963) 335.[7 ] STOKER, J .J . , D iffe ren tia l G eom etry , W iley -In te rsc ien c e , New York (1969).

IAEA-SM E-U/32

PHASE TRANSITIONS IN D-DIMENSIONAL ISING LATTICES

R .A . GOLDSTEIN Department of M athem atics,

J.J. KOZAKDepartment of Chemistry,U niversity of Notre D am e, Notre D am e,Ind. , United States o f Am erica

Abstract

PHASE TRANSITIONS IN D-DIMENSIONAL ISING LATTICES.U nder th e assum ption o f its ex isten ce , th e p a rtitio n function o f a D -d im ensiona l Ising la t t ic e is

e v a lu a te d in various reg im es o f 6 = 1 /k T , and possib le phase transitions are ch a ra c te r iz e d .

1. INTRODUCTION

The fundam ental problem of m ech an ics is that of c h arac te r iz in g the tim e evolution (or flow) of a dynam ical sy stem given the in itia l conditions. R e c a ll that a dynam ical sy stem is a co llection (M,ju,cpt ), w here M is a sm ooth m anifold , ц a m e a su re on M defined by a continuous p o sitive density, and the "flow " eft : M -» M (t e IR) i s a o n e-p aram eter group of m e a su re - p re se rv in g d iffeo m o rph ism s [1]. (L o cally , cpt i s defined by a f ir s t - o r d e r sy stem : x =f(x). ) If the sy stem contains re la tiv e ly few p a r t ic le s , then a com plete d e scrip tio n in te rm s of the H am iltonian equations of m otion

q =9HЭр

ЭН9q

w here (q, p ) e R 6N, N is the num ber of p a r t ic le s and H = H(q, p) i s the H am iltonian, i s , in p rin c ip le , p o ss ib le , and the flow can be determ ined by in tegratin g the d iffe ren tia l equations given the in itia l co -o rd in a te s and m om enta of the individual p a r t ic le s . On the other hand, if the sy stem contains v e ry m any p a r t ic le s ( e .g . A v o gad ro 's num ber), then the above approach is re a lly out of the question , and, a s re a liz e d by M axw ell, B o ltzm ann , G ibbs and P o in caré , it i s f a r m ore sen sib le to introduce a s ta t is t ic a l approach fo r the study of com plex dynam ical sy s te m s . W hereas the s ta t is t ic a l approach would seem to be fo rced upon u s by our ignorance of the in itia l co -o rd in a te s and m om enta of 1023 p a r t ic le s , and the im p o s s ib ility of com puting sim u ltan eo u sly the dynam ical h isto ry fo r each of th ese p a r t ic le s , it i s w ell to keep in m ind the "p o in t" of doing such a calculation . We a re rem inded of S ch rô d in g er 's often-quoted re m ark [2]: "I t dawns upon

245

246 GOLDSTEIN AND KOZAK

u s that the individual c a se is en tire ly devoid of in te re st , whether detailed in form ation about it i s av a ilab le o r not, whether the m ath em atica l problem can be coped with or not. We re a liz e that even if it could be done, we should have to follow up thousands of individual c a s e s , and could m ake no b ette r u se of them than compound them into one s ta t is t ic a l enunciation.The w orking of the s ta t is t ic a l m ech an ism it se lf i s what we a re re a lly in te re sted in ". T h is sh ift from a d e te rm in istic to a p ro b a b ilis t ic point of view in the study of com plex dynam ical sy ste m s c a r r ie s with it the im p lication that a contracted d escrip tio n of a m any-body sy stem , one that involves few er than the 6N co-ord in ate and p osition v a r ia b le s , i s not only n e c e ssa ry but d e s ira b le . That such a contracted d escrip tio n , one in which a new se t of v a r ia b le s is identified, can be constructed is , p erh ap s, not too su rp r is in g . What i s aston ish in g , how ever, i s that the entire body of in form ation fo r a sy ste m of 1023 p a r t ic le s can be su m m arized in te rm s of three o r four v a r ia b le s (depending on the co n stra in ts im posed and the com position of the system )'. T h is contraction i s known a s th erm odyn am ics, and it m ay be taken a s the p rin c ip a l aim of s ta t is t ic a l m ech an ics to attem pt to show how it i s that the m ac ro sc o p ic behaviour of m atte r , a s sy stem ized in the law s of th erm odyn am ics, fo llow s from (or can be understood in te rm s of) the m ic ro sc o p ic law s of atom ic theory.

In recen t y e a r s , an in cre a sin g num ber of m ath em atic ian s and ph ysical s c ie n t is ts have turned th eir attention to an exam ination of the m ath em atical s tru c tu re and underlying a ssu m p tio n s of equ ilibrium and nonequilibrium s ta t is t ic a l m ech an ics fo r the e x c e lle n t re a so n that there s t i l l rem ain fundam ental p ro b lem s which a re , a s yet, unsolved, th is desp ite the fac t that ce rta in of th ese p ro b lem s have been studied active ly fo r over a century. T o p lace th ese p ro b lem s in th e ir p ro p er context, and to sta te the m ath em a tic a l q uestion s involved p re c ise ly , it would, of c o u rse , be n e c e s sa ry to undertake a com plete exposition of the theory, a ta sk which is obviously u n re a lis t ic given the sp ace lim ita tio n s of th is contribution. R ath er, what we sh a ll try to do h ere , i s to pose one problem of cu rren t in te re st in s ta t is t ic a l m ech an ics in som e d etail, the problem of ph ase tran sitio n s in D -d im en sion al Isin g la t t ic e s , and then to indicate how a sy n th esis of b ifu rcatio n theory and the theory of a sy m p to tics can lead to a p re lim in ary understan ding of the o ccu rren ce of s in g u la r itie s in the therm odynam ic functions at certa in , w ell-defined points in the tem peratu re spectrum .

We m ention in p a ss in g that a fundam ental p roblem in s ta t is t ic a l m ech an ic s, a problem which we sh a ll not even co n sid er in th is p ap er, is the m ath em atica l p roof of the ex isten ce of a certa in function which p lay s an e s se n t ia l ro le in the theory. T h is function, ca lled the p artitio n function Q, i s understood a s (ap art from a n orm alization constant) the sum over a ll en ergy s ta te s Ej of the sy stem under study,

Q exp [ -/3 Ej ]j

w here 3 i s a p a ra m e te r re la ted to the tem peratu re v ia B o ltzm an n 's constant, к (in p a r t ic u la r , 3 = 1 /kT ). It tu rn s out that a ll m ac ro sc o p ic therm odynam ic v a r ia b le s (e. g. p r e s s u r e , fre e energy , entropy, etc. ) can be e x p re sse d in te rm s of th is function, and when p h y sic is ts o r ch em ists "d o " s ta t is t ic a l m ech an ic s, a p r im a ry ob jective i s to evaluate Q, given the c la s s ic a l o r

IAEA-SMR-11/32 247

quantum -m echan ical law s d e sc r ib in g the m o lecu lar m otion and in teraction s of the p a r t ic le s co m p risin g the sy stem being studied. It i s in terestin g to note, then, that the fundam ental assu m ption of B o ltzm ann and G ibbs on which the ex isten ce of the partitio n function r e s t s , the so -c a lle d ergod ic h ypoth esis, w as proved only recen tly by Sinai [3] fo r a sy stem of N 3 h ard sp h e re s (that is , p a r t ic le s in teractin g v ia in fin itely rep u ls iv e fo rc e s fo r r < a, w here a i s the d iam eter of the p a r t ic le , and experien cin g no fo r c e s beyond a) m oving in a volum e V with p erio d ic boundary conditions.A p roof that sy s te m s whose m o lecu le s in teract v ia rep u lsiv e and a ttractiv e fo r c e s (i. e. sy s te m s in the " r e a l w orld") a re a lso m e tr ic a lly tran sitiv e h a s not yet been achieved! With re sp e c t to the p re sen t d isc u ss io n , and fo r the c la s s of sy s te m s co n sid ered in th is p ap er, we sh a ll a ssu m e the ex isten ce of the p artitio n function, with our m ore m o d est ob jective being the evaluation (in som e sen se) of the p artitio n function in d ifferen t re g im e s of /3, and the c h arac te r iza tio n of p o ss ib le (seco n d -o rd er) ph ase tran sitio n s in D -d im en sion al Isin g la t t ic e s .

C o n sid er a la tt ice with a sp in at each v ertex of the la ttice , and a ssu m e that each sp in a can take on two v a lu e s , +1 (sp in u p ) and -1 (sp in down). C le a r ly , if there a re N v e r t ic e s , there a re a to tal of 2N p o ss ib le configu ra tio n s of sp in s on the la ttice ; le t u s denote a p a r t ic u la r configuration of th ese N sp in s by { a } . In the sp e c ia l c a se with which we sh a ll be concerned in th is p ap er, i. e. the Isin g m odel, the en ergy fo r one p a rt ic u la r configuration of sp in s is

the p rim e on the f i r s t sum indicating that the sum is over n e a re st neighbours only. H ere , J i s the "cou p lin g constant" betw een neighbouring sp in s , and H is the ex tern a l m agn etic fie ld . T h us, the f i r s t sum on the right-hand sid e accounts fo r sp in -sp in in teractio n s and the second term re p re se n ts the in teraction betw een the individual sp in s and the ex tern a l m agn etic fie ld .In one dim ension , the la ttice b eco m es a chain of N sp in s, and

fo r a la ttice with m row s and n colum ns, etc. We re m ark that the su b sequent a n a ly s is i s fac ilita te d if one adopts p erio d ic boundary conditions. A ccord in gly , in one d im ension the ends of the open chain a re linked to form a rin g ; in two d im en sio n s, the la ttice i s m apped on a to ru s , and in h igh er d im en sio n s, s im ila r m appin gs of the la ttice a re con structed .

i.J

N -1 N

w hile,in two d im en sion s

E { a } = - Ji= i j= i 1=1j= i 1=1 j=l


In p h y sica l te rm s , we note that sin ce the in teraction en ergy between two neighbouring sp in s i s - J if both sp in s a re up (or both down), and + J if one spin i s up and the other down, one obtains a low er en ergy fo r a p a r a lle l configuration if J > 0. T h e re fo re , the m inim um en ergy fo r the p a r a l le l configuration is ju s t

where q in th is e x p re ss io n is the num ber of n earest-n eigh b ou r s it e s of a given s ite . W hereas in the p re sen ce of a m agn etic fie ld H > 0 the m inim um en ergy i s achieved fo r a ll sp in s p a ra lle l and up (with ju st the opposite configuration fo r H < 0 ) , in the ab sen ce of a m agn etic fie ld (H =0) the m inim um en ergy i s achieved e ith er when a ll sp in s a re up or a ll sp in s are down. The situation J > 0 co rre sp o n d s to fe rro m ag n e tism , while J < 0 c o rre sp o n d s to an tife rro m agn etism (here one obtains a low er en ergy fo r an tip ara lle l con figuration s). F in a lly , we note the ex isten ce of a quite d ifferen t p h y sica l in terpretation of the Isin g m odel, nam ely, the lattice g a s ; h ere one a s s ig n s o. = +1 if a m olecu le i s at the ith la ttice s ite , and CTj = -1 if there i s no m olecu le at the ith la ttice s ite . T h is la t t ic e - g a s in terp re tatio n is quite u se fu l s in ce it a llow s the con sid eration of g as- liq u id e q u ilib r ia , and the study of phenom ena n ear the c r it ic a l point.

The s ta t is t ic a l-m e c h a n ic a l problem can now be posed : Given the en ergy E { ct} fo r a p a r t ic u la r la ttice , evaluate the partitio n function

w here the sum ov er {a } denotes a sum over a ll p o ss ib le configurations cfj = ±1 of N sp in s on the la ttice . Now, what h a s been proved fo r th is p ro b lem ? The on e-d im en sion al p rob lem h as been so lved both in the p resen ce and ab sen ce of a m agn etic fie ld [4]. The tw o-d im ensional problem h as been so lved only in the ze ro -fie ld c a se ; th is , of c o u rse , i s the fam ous contribution o f O n sager [5]. Nothing w hatever i s known about the exact m ath em atica l solution to the th ree-d im en sio n al problem , e ith er in the p re sen ce or ab sen ce of a m agnetic fie ld . Our sp ec ific concern in th is p ap er is to in dicate an approxim ate m ethod fo r obtaining the behaviour o f the D- d im en sion al Ising m odel in the ze ro -fie ld c a se .

2. THE Z E R O -F IE L D ISING M ODEL

C o n sid er a r e g u la r la tt ice with a spin at each of the N s it e s and a ssu m e that the in teraction en ergy betw een neighbouring sp in s m ay be w ritten a s - J ct. cr. where J ( J ^.0) i s the in teraction energy , and each spin can take on the d isc re te v a lu e s ±1. Then, in the ab sen ce of a m agn etic fie ld , the p artitio n function can be w ritten down at once; it is

E 0 = -JN q/2 -N |H ¡ (period ic la ttice s)

exp -kT

(1)

{ ° i = ±1}

IAEA-SMR-11/32 249

H ere, the se t (a } denotes a p a r t ic u la r configuration of sp in s , and S ' denotes a sum over n e a re st neighbours (only); the fa c to r 2"N i s included so that the p artitio n function is n o rm alized to unity. Suppose now that in com puting the sum o v er n e a re st neighbours, we count each in teraction tw ice; then, u sin g the notation 2 to indicate th is double counting, the partitio n function a s su m e s the fo rm :l,)

Qn = 2~N У

{°i = ±1}

exp 0 J(2 )

The point of counting each in teraction tw ice i s that the m a tr ix correspon din g to the q u ad ratic fo rm i s sy m m etric . B e c a u se of th is sy m m etry , a

i.jd iagon alization of the q u adratic form is alw ays p o ss ib le (via an orthogonal tran sfo rm atio n ), and if, in addition, we a ssu m e p erio d ic boundary conditions then the m atr ix of the co e ffic ien ts w ill be of the cy c lic fo rm . F o r d e fin iten e s s , le t 3 denote the colum n vector:

B e r l in and K ac [6] show exp lic itly that a m atr ix M can be con structed such that N

^ a¡ о- = 5 r>l- M ' < / = î.j i. j = 1

ai Mij ffj

w here the cy c lic m atr ix M h as the r e a l , orthogonal e igen v ecto rs V (n orm alized to unity):

(3)

Vk = v ks cos 2ir ( k - l ) ( s - l ) N

27r ( k - l ) ( s - l ) N (4)

and e igen valu es m p

D

Im p = 2 ) co su=l

27г р „ p =(p 1,p2,...,p d)0 S p â L - 1

(5)

T h us, the Isin g p artitio n function can be w ritten in the form :

Q = 2"n У e x p | ^ a ' • M • a j-

h = ±1}

where H = j3J.

(6)

2 50 GOLDSTEIN AND KOZAK

A second sim p lifica tio n in the stru c tu re of QN can be introduced if one u s e s the follow ing, lin e a r a lg e b ra ic gen era lizatio n of the G a u ssian in te g r a l[7]:

exp' 2 ^ k J ki

k,{

= (2тг)~2 (det J exp ■| Ук ( •Т ’ 1)м У ^ + i k k.fi к

П<

(7)

T h is identity i s valid fo r any se t of n ( re a l o r com plex) v a r ia b le s Çk and fo r any r e a l , sy m m etric p o sitiv e-d efin ite m a tr ix J . Indeed, th is identify could be u sed im m ediate ly w ere it not fo r the fac t that the cy c lic m atr ix M need not be p o sitiv e definite (the e igen valu es of M a re co sin es) though the m atr ix i s r e a l and sy m m etric . H ow ever, th is d ifficu lty i s handled e a s i ly be adding, and then su b tractin g , a quantity a(a> 2D) to the d iagonal e lem en ts of M. S p e c ifica lly , we w rite

M = M + al -al

w here I i s the identity m a tr ix , and then define

W = M + a I

(8)

(9)

The m a tr ix W is now a re a l , sy m m etric , p o sitive definite m atr ix with e igen valu es

“ p = m p + a (10 )

With the above id en tificatio n s, the G a u ssian identity can be u sed d irec tly (se t k r e a l) , and one obtains

NHЧ , = 2 Г « е ^ “ I 6XP

{o¡ = ± 1}

H W • a

- \ >) (2тг)‘ 2 (det HW)-i

exp y 1 • (HW)-

N1 —1 *y exp 1

l> Q SC & dNy ( И )

k=l

1Since the m atr ix W ap p e a rs everyw here a s W 1 it i s convenient to identify a new m atr ix

—» —» A = W" (12)

with e igen valu es

2 i r p vL LÏ .

P = (Pp P2< • • • - Pd) <13)0 - P r t f - 1

V = 1

IAEA-SMR-11/32 251

Then, a fte r som e m an ipu lation s, one obtains

Qn = exp.n 4 , Г +°° P

(2тгН) 2 (d e tA )2 J . . . J exp { I(y)} dNy (14)

where

I(y) = - ^ | < y , A y > + ^ log cosh ÿ (14a)

3. A SY M PTO TICS

The d ifficu ltie s a sso c ia te d with evaluating QN exactly lead one to co n sid er the p o ss ib ility of obtaining an asym ptotic e stim ate fo r the partition function. L e t u s focu s on the in tegra l appearin g in the e x p re ss io n fo r Q¡ :

+ «

S d Ny exp{I(y)} (15)

and attem pt to con struct

log SwL im — N N - S (16)

M ore p re c ise ly , we e stim ate SN by the m ethod of s te e p e st descen t, w here, in th is problem , H appearin g in the in tegra l Sn p lay s the ro le of the la r g e n e s s p a ra m e tr [8].

In our approach , we begin by con sid erin g the behaviour of the in tegra l, E q . (15), in the neighbourhood of each of it s c r it ic a l points. We expand the in tegrand in a T a y lo r s e r ie s about each of it s c r it ic a l points, and then sum up the contribution from each of those points. About one such c r it ic a l point, sa y yC]., we w rite

Ky) =I(yCj) + | < S y Cj, V2I(yCj) 6yCj > + . . . (17)

where

6yCj = У - yCj

Note that sin ce yCj i s a c r it ic a l point, V I(yc.) = 0. T h e re fo re , the con tribution to the in tegra l I(y) at the c r it ic a l point yc. is e stim ated to be

yCj+A

J e x p |l (y c.) + |- < 6y c.,V 2 I(yc.) 6yc. > | d Ny (18)

yC j- ¿

2 5 2 GOLDSTEIN AND KOZAK

w here Д i s a sm a ll neighbourhood about yCj.. If one p ro ce e d s to sum up the contribution from each of the other j c r it ic a l points (j = 1 , 2 , . . . , s), the net re su lt is

+ 00 b +00

J. . .J exp{I(y)} dNy - If . . ./ e x p |l (y c.) + | < 6yc. , V2I(yc. ) 6yCj > | dNy-00 j = 1 “00

(19)

w here, re la tiv e to each c r it ic a l point, we have extended the ran ge of in tegration to ( -00, +co), sin ce the overw helm ing contribution to the in tegra l w ill com e fro m the narrow ran ge (yc. - Д, yc. + Д), with a neglig ib le contribu tion from anything outside th is range.

Now, the m ethod of s te e p e st descen t w orks fo r m axim um points, but in our approach we focu s on c r it ic a l points in itia lly . Given th ese c r it ic a l po in ts, we p roceed by tran sfo rm in g the path of in tegration into one of s te e p e st d escen t, such that the c r it ic a l points becom e m axim um points. M ore exp lic itly , we tran sfo rm the in tegra l, Eq . (19), to a com plex in tegra l in the com plex N -plane, by u sin g C au ch y 's in teg ra l fo rm u la . C on sid er the follow ing co-ord in ate tran sform ation :

y j _> giCïj (y j - yi . )

We then choose r e a l co e ffic ien ts ttj (j = 1 , 2 , . . . ,N ) such that V l (y )< 0 , a condition which g iv e s the m axim um points. The consequence i s that one ob tain s the sam e re su lt with yc <-» y 1 =0 co rrespon din g to a m axim um point on the path of s te e p e st descen t.

Once the above change of co -o rd in a te s h a s been m ad e , we change co o rd in ates once aga in so a s to d iagonalize V2I(yc), and a fte r p erform in g som e ca lcu la tio n s, we obtain the follow ing e stim ate fo r the free energy:

-F(H) = lim = lim -i- log j expN->«> iN N -*■“ iN

- - N * H H (det A ) ^ф И ( У с , ) ]

j=i{det[ -V I(yc )]}’

+ 0 ( e ' a /H ) (H -0 ) (2 0 )

4. DETERM INATIO N OF THE C R ITIC A L PO INTS

H aving obtained an e stim ate fo r the partitio n function, E q . (20), we m u st now determ in e the se t of c r it ic a l points of the function I(y). The c r it ic a l points of I(y) a re the points fo r which

Э1(У)3y‘

= 0 (i = 1 ,2 , . . . ,N ) (21 )

y‘ =y¿

or,

У Ai] y J + tanh y 1 = 0

j=i У1 = Ус

(22 )

T akin g into account a ll of the y 1, we obtain the o p era to r equation

A yc = H 6 (yc) (23)

IAEA-SMR-11/32 253

w here,— — - -

Ус1 tanh y¿

Ус = Ус 6(УС) = tanh у I

Усы tanh y^

Since tanh(O) = 0, we see at once that a p a r t ic u la r solution to E q . (23) fo r a ll H is

Уо

w here y0 i s an N -com ponent v ec to r with one ze ro correspon din g to each y ‘ (i = 1 , 2 , . . . ,N ). In fac t, one can prove the follow ing theorem :T h eorem : F o r H < Aj = l / ( a + 2D), the m inim um eigenvalue of A, the only c r it ic a l point of I(y) i s y0 = 0.

H aving identified the sin g le c r it ic a l point re la tiv e to the ran ge H < Xj we now seek to determ in e the se t of c r it ic a l points fo r H >X j. T h is am ounts to finding the other so lu tion s of E q . (23); that i s , we determ ine whether new solutions m ight a r i s e from the known solution y0 fo r p a r tic u la r v a lu e s of the p a ra m e te r H. P h ra se d d ifferen tly , we study the p o ss ib ility that new so lu tion s can b ifu rcate from the known solution y0. We r e c a ll that if b ifu rcatio n tak e s p la ce , it m u st happen at one of the e igen valu es of the o p era to r A. A ccordin gly , we w rite the gen era l c r it ic a l point yCv in a pow er s e r ie s in a p a ra m e te r e about the known c r it ic a l point y0, and, s im ila r ly , we expand the stren gth p a ra m e te r H in a pow er s e r ie s about the value H0 (the la tte r to be determ ined). The perturbation schem e u tilized below g o es back to E u le r , P o in caré and R ellich [9]

Ус„ = Уо+ e У1 + е 2У2 + е3Уз+ • • •

Н = Но + е Hi + е2 Н2 + е3 Н3 -

(24)

(25)

We p roceed by substitu tin g th ese expan sion s into the o p era to r equation

A y c„ - H tanh yCy= F (e )


and then, d ifferen tia tin g F (e ) su c c e s s iv e ly with re sp e c t to e , we se t the r e su lt at each o rd e r to zero . S tartin g off, we w rite

9F(e)de

from which we determ in e the re la tion

= 0 (27)e = 0

A y1 = [H sech2 yc • Э6 yc + H£ tanh yc]£ = 0 (28)

T h is le a d s to the re su lt

А У1 = H0 ya (29)

T h is equation, which is ju s t the v aria tio n a l equation correspon din g to E q . (23), d efin es a lin e a r eigenvalue problem . We identify

h 0 = x „ (x1 a 2 s . . . a B s . . . a N) (30)

У1 = А „Ф „ (31)

w here the cp„ a re the e igen v ecto rs correspon din g to the e igen valu es X of the o p era to r A and A v i s som e constant, to be determ ined la te r . T o continue, we con struct

9 ^ ( 6Эе2

= 0 (32)e = 0

which le a d s to the re su lt

( A - X J y2 = 2 H j A v Ф„ (33)

T h is equation h a s a solution if and only if the righ t-hand sid e is orthogonal to the null sp ace of (A -X„), i. e. to the eigenfunctions of A correspon din g to X„. B u t, the eigenfunctions of the hom ogeneous equation a re (to within a constant) ju s t ф„, i. e.

(34)

H ence, the righ t-hand sid e of E q . (33) can never be orthogonal to the фи u n le ss Hj = 0. A ccord in gly , from the second varia tion , we have

H j = 0 (35)

У1 = A V %

У2 = c„ A„ % (36)

IAEA-SMR-11/32 255

w here cv i s som e constant. F in a lly , we con struct

93F(e)Эе3

€ = 0which le a d s to the re su lt

w here,

and where

( A - X J y 3 = ЗНзА^Ф^-гХ^ А3 Ф

(37)

(38)

(€ )3 0 0

Ф = 0 (ф ' )3 . . . 0 (39)

0 0 Ф 3

2TTÍV 2t i v 1(40)% = CO S N + s i n . N J

i s the i-th com ponent of the у-th e igen vector of A. G iven the stru c tu re of the correspon din g hom ogeneous equation, E q . (38) h a s a solution iff the righ t-hand sid e i s orthogonal to фу. T h is le a d s to the re su lt

(41)

w here,N

I № ID„ = 2 1=1 (42)

i = 1

Given the eigenfunctions, E q . (40), the coeffic ien t Dy can be evaluated , and one fin ds that D„ = 1 . H ence.

Ho = X„ At (43)

Substitution of the r e su lt s obtained thus f a r into the perturbation expansions fo r y„ and H le a d s to

Ус„ = Ф„(е+е2) + 0 ( е3)

H = X1/(l + e2) + O ie 3)

(44)

(45)

w here we have u sed that yQ = 0 and HQ = and taken advantage of the lib e rty in ch oosing cv to se t cv = A v; fin ally , fo r s im p lic ity , we rep laced A ye by e.


R elative to the ran ge H < Xj, we had proved that there e x is t s only one c r it ic a l point of I(y), nam ely, y0 = 0. One can show quite e a s i ly that y0 = 0 i s a lso a m axim um point; th is fo llow s fro m the fac t that the m atrix ^ 2 *(Уо) can be shown to be negative definite fo r the ran ge H < X1#

One can p ro ceed , then, with the evaluation of the p artitio n function, E q .(2 0 ), and determ in e an e x p re ss io n fo r the fre e en ergy of the sy stem v ia the re la tio n

-F = lim * ° g Q n (46N-»oO

5. THE F R E E EN ER G Y IN TH E RANGE H <

The re su lt fo r the fr e e en ergy is :

2тг

F = -2 ( 2 т г ) 1 \

. . . / dDu log

D

1 -2H ^ co s u1

i = l(H -0 ) (47)

where in the continuum lim it we have le t o'-*-2D and rep laced Xx by A(w) where

X(w) =■ D

' I0 § u S 2 i (48)

2 ) c o s u ‘ i = l

We see that the asym ptotic solution re la tiv e to the ran ge H < х г (H -> 0) i s exactly the G a u ssian m odel of B e r lin and K ac [6]. Our approach y ie ld s that in the lim it N -»°o, H ->0, the Isin g m odel should behave like the corresp o n d in g D -d im en sion al G a u ssian m odel.

6. THE F R E E EN ER G Y IN THE RANGE H > XN

In the ran ge H >XN, it tu rn s out that V2 I(y0) > 0 ; in other w ords, y0 is a m inim um point fo r I(y) in th is ran ge. T h u s, the G a u ssian m odel y ie ld s the c o rre c t lim itin g behaviour of the Isin g m odel fo r H <X 1, but f a i l s to re p re se n t the Isin g m odel in the ran ge H > XN . It i s , how ever, a m odel "b ifu rca tin g fro m " the G a u ssian m odel that y ie ld s the c o rre c t behaviour in the ran ge H >X N. U sin g b ifu rcatio n theory, we can determ ine not only w hether b ifu rcatio n w ill occu r from the known solution yQ = 0, but a lso we can e stim ate the stru c tu re of the new solu tions which sp rin g from y0 = 0 at H = XN.

We turn our attention to the evaluation of the determ inant

-V2I(yc„) = | - sech2 (yCu) (49)

T h is i s achieved by determ in in g the e igen valu es Uj and e igen v ecto rs r¡¡ of the m atr ix H S , where

H S (yCl/) = A - H sech2 (yc¡1) (50)

IAEA-SMR-11/32 257

We write

Uj (e) =

k = 0

£ _ k k !

к = 0

Substituting th ese e x p re ss io n s into the o p era to r equation

H <g Uj = rjj Uj

and m aking u se of the lim its

(51)

(52)

( 53 )

lim H = X„0

ЭН „ l u n — = 0 e-> о de

f í = 2Xy

lim Yr =0e-» 0 v

Ятг.»lime-*0 9 e

Я2lim —b i = 2ф„eô Эе

one fin ds, u sin g the sam e proced u re a s the one introduced in sectio n 4, that the e igen valu es of the determ inant H [ -V2I(yc. )] a re given by

>1j (e) = (Xj-X„) + X„ e2[ L „ j -1] (54)

<55)

where

H ere, the cp¡ a re the eigenfunctions and the Xj a re the e igen valu es of the o p era to r A, i. e.

A cpj = Xj ф.

F u rth e rm o re , we find that, to o rd e r e3,

N Nf t т = е 2 [ Ьу>„ - 1 ] Д Xj П ( l - Y - ) + 0 ( e 3 ) ( 56)

j = l j = l \ * v V i y

With the re su lt , E q . (57), the p artitio n function in the ran ge H > Xj a ssu m e s the form :

D

2 * j yQ n

N D

- H4 e x p |l(y Cw) l o g 1 - Xy a + 2 ^T co s } (57>" i * V Li = l

X exp - — NaH


We note that th is e x p re ss io n fo r can be w ritten in the a ltern ate form :

( a - 2 D )-1

Qn = O(N) J àX exp {N E (X)} (58)(a+ 2D )_1

withD

~J dDu log 1 - x (a + 2 ^ c o s u yE (x > = - | * l s + 1 ^ - | l 0s (2X - H) - ¿ Д > .

i=1 (59)where we have rep laced the sum L by an in tegra l over u, and rep laced

j * V

the sum over X by an in tegra l over X. T h is rep resen ta tio n is achieved by app lication of C au ch y 's in teg ra l fo rm u la and the re sid u e theorem . F in a lly , the contour of in tegration w as deform ed such that the ran ge of X jis (X , oo).

Upon exam ining the above e x p re ss io n s (E q s (58, 59)), we note that the in teg ra l in the exponent h a s a branch point at X =Xj w here Xj =1 /2D , and it i s at th is point w here (ex ac tly a s B e r lin and K ac show fo r the sp h e rica l m odel) that the p o ss ib le ph ase tran sitio n tak e s p lace . Note, how ever, that we m u st s t i l l determ ine the value of H at which the tran sitio n o ccu rs. A ccord in gly , in the next sectio n we take up the problem of ch arac te riz in g the p o ss ib le s in g u la r it ie s in the fre e energy , where the fre e en ergy is determ ined from the re la tio n , E q . (46). We proceed by u sin g , once again , the m ethod of ste e p e st descen t (though now in it s c la s s ic a l version ).

7. THERMODYNAMIC BEHAVIOUR IN THE RANGE H È 1/2 D, AND PH ASE TRANSITIONS

T o evaluate the in tegra l, Eq . (58), by the m ethod of s te e p e st descent, we need the m axim um points of E(X); we r e c a ll that our a n a ly s is based on b ifu rcatio n theory gave only c r it ic a l poin ts, but the co-ord in ate t r a n s form ation su ggested in sectio n 3 allow s the identification of the m axim um poin ts on the path of s te e p e st descen t. L et u s denote by Xs = XS(H) the pathon which we a re a s su re d that E '(X$) =0 , o r, equivalently,

é ' ¿ T r - ¿ K<2X> <60>

while at the sam e tim e E"(Xs ) < 0 , o r equivalently,

E " (X‘ ) = ( 2X T ^ - è K(2X) + ¿ K ' ( 2 X ) < 0 (61)

w here, in the above e x p re ss io n s ,

12l\

K(2X) = (2 I dDy ---------- iS--------- (62)0

D

l - \ ( a + 2i= 1

V) C O S Ü J

IAEA-SMR-11/32 259

We re m a rk that the Xs = XS(H) which sa t i s fy E q s (60) and (61) a re the m axim um points to be u sed in the calcu lation of the fre e en ergy v ia E q . (46). F o r the p u rp o se s of the p re sen t d isc u ss io n , we exam ine the behaviour of the heat cap ac ity

C„ = к H2( ^ | ) (63)\ эн2/ эх Чэн/ ax2

on the path X = XS(H) to determ in e whether C v exh ib its a sin gu la rity , and if so , we sh a ll c h a ra c te r iz e the behaviour in the neighbourhood of the sin g u la rity , and sh a ll locate the point at which the ph ase tran sitio n tak e s p lace .

In one d im ension , E(XS) h as a "p o ten tia l" sin g u la r ity when H->-§ and X —> H ow ever, by con siderin g the behaviour of E"(X S) in the neighbourhood of the sadd le point Xs , we find that

lim E " (X ) = +00

i

T h at i s , the point ( i , \) d o es not lie on the sad d le curve [H,XS(H)]. H ence, we conclude that the fre e en ergy and a ll d eriv ative p ro p e rtie s a re w ell behaved throughout the ran ge H > X^, th is b ecau se the "p o ten tia l" sin gu la rity at H = •§-, X = ^ c o rre sp o n d s to a m inim um of the in tegrand, and, con sequently , con tribu tes n eglig ib ly to the therm odynam ic behaviour of the sy ste m . It i s in terestin g to note that although we do have b ifu rcatio n in one dim ension , we do not have a ph ase tran sitio n [ 10].

In two d im en sio n s, exam ination of the stru c tu re of K(2X) show s that the in teg ra l m ay have a sin gu larity when 2X = | and a l l u ^ O . A gain , we exam ined the behaviour of E "(X S) in the neighbourhood of the p o ssib le sin gu la rity at H = I , X = 1 to determ ine whether the point ( i , 1 ) l i e s on the sadd le curve [H ,XS(H)]. We find that

lim E "(X S) = -ooH ^ iX - i

In other w ords, a sin g u la r ity in the therm odynam ic heat cap ac ity i s ach ieved in two d im en sio n s, s in ce E"(X S) < 0 co rre sp o n d s to a m axim um of the in tegra l. M o reov er, the behaviour of Cv in the neighbourhood of the sin gu la rity i s found to be logarith m ic in agreem en t with the behaviour found by O n sager in h is exact solution of the tw o-d im ensional Isin g m odel. Note that w h ereas the sin gu la rity in X o c c u rs at the b ifurcation point,X = ^ , the sin gu la rity in H i s sh ifted beyond the b ifu rcatio n point. The heat cap ac ity i s w ell behaved until the strength p a ra m e te r H ach ieves a value of 0.25) a value which a g r e e s fav ou rab ly with the O n sager re su lt ,Hc = 0.44.

C o n sid erin g next the c a se of three d im en sio n s, we again exam ine the sin g u la r ity stru c tu re of E '(X ) =0 along the sad d le curve, Xs =XS(H). One p o ss ib le s in g u la r ity i s the in tersec tio n of Xs = XS(H) with the line X= 1/2 H;

26 0 GOLDSTEIN AND KOZAK

th is in tersec tio n is found to occu r at X = H = 0 , and hence at infinite tem p eratu re - a point which is never reach ed . The second p o ssib le s in g u la r point is at X = — , a point where the in tegra l appearin g in E q . (59) m ay becom e sin gu la r . H ere we find that

lim E "(X S) = -ooH-» 0.12

K 12

T hat is , a sin g u la r ity in the heat cap ac ity i s achieved when Xs -» 2 and H -» Hc =0 . 12 sin ce E "(X s) < 0 . We note that the nature of the sin gu larity in the neighbourhood of the point (HC,XC) = (0 .12 , ^ ) i s a lg e b ra ic , behaving like (6 X - I ) '1/.2 What we have not shown is the p re c ise behaviour of Cv with re sp e c t to (H - Hc) (which am ounts to so lv in g fo r \ = XS(H)), and th e re fo re we have not determ ined the c r it ic a l exponent a. H ow ever, by cons id e r in g XHh one show s in exactly the sam e m anner a s B e r lin and K ac, that XHH h as a jum p discontinuity , that i s , there e x is t s a d iscontinuity in the slope of Cv at Hc. F in a lly , it i s worth m entioning that the n u m erically e stim ated value of Hc fo r the th ree-d im en sio n al Isin g m odel i s 0 .22, and our c r it ic a l tem p eratu re Hc = 0.12 i s in fa ir agreem en t with th is e stim ate [1 1 ].

We m ention in c lo sin g that an an a ly s is s im ila r in sp ir it to the one p resen ted above im p lie s that fo r D s 4, the in teg ra l K(2XS) beh aves like

(X -XC)(D 1^ 2 log(X -X c) even d im ensions

(X -X CP 2 2 odd d im ensions

Inspection of th ese re su lt s r e v e a ls that fo r D è 4 there i s no lon ger any sin g u la r ity in the sp e c ific heat,but the d iscontinuity in the slop e of the sp ec ific heat s t i l l p e r s i s t s . F o r d e ta ils on the above ca lcu latio n s, the re a d e r m ay consu lt R ef. [12].

A C K N O W L E D G E M E N T S

The g en era l approach taken h ere w as su ggested to the auth ors by P r o fe s s o r R obert W. Zw anzig of the U n iv ersity of M aryland. The authors w ish to acknow ledge se v e ra l v e ry helpful d isc u ss io n s with P r o fe s s o r Zw anzig.

R E F E R E N C E S

[ 1] ARNOLD, V. I . , AVEZ, A ., Ergodic Problems of C lassical M echanics, W .A. Benjamin, In c ., New York (1968).

[2 ] SCHRÔDINGER, E. , Nature 153 (1944) 704.[3 ] SINAI, J a .G ., in S tatistical Mechanics: Foundations and Applications, W .A. Benjamin, In c .,

New York (1967).[4 ] See, e . g . , THOMPSON, C .J . , M athem atical Statistical M echanics, The M acm illan Company,

New York (1972).[5 ] ONSAGER, L . , Phys. Rev. 65 (1944) 117.

IAEA-SMR-11/32 261

[6] BERLIN, Т .Н ., KAC, М . , Phys. Rev. 86 (1952) 821.[7] The continuum representation of the D-dim ensional Ising model has been used by many authors.

See, e .g . LANGER, J .S . , Ann. Phys. 41 (1967) 108.[8 ] For a discussion of this type of integral, see: DE BRUIJN, N .G ., Asymptotic Methods in Analysis,

North Holland, Amsterdam (1961).[9 ] For a recent monograph on applications of bifurcation theory, see: KELLER, J .B ., ANTMAN, S .,

Bifurcation Theory and Nonlinear Eigenvalue Problems, W. A. Benjamin, I n c . , New York (1968).[10] The absence of a correlation between bifurcation points and phase transitions in one dimension was

also found in a study of the Kirkwood-Salsberg hierarchy of integral equations for a system of hard rods. See; ING-YIHS. CHENG, KOZAK, J . J . , J. Math. Phys. 14 (1973) 632.

[11] See, e .g . FISHER, М ., Rept. Progr. Phys. 30 (1967) 615.[12] GOLDSTEIN, R. , KOZAK, J. , Physica 71 (1974) 267.

IAEA- SM-11/33

DIFFERENTIAL CALCULUS IN LOCALLY CONVEX SPACES

R. A. GRAFFDepartment of Mathematics,University of California,Berkeley, Calif.,United States of America

Abstract

DIFFERENTIAL CALCULUS IN LOCALLY CONVEX SPACES.A class of locally convex spaces which the author calls d ifferential spaces (D-spaces) is defined. The

class includes a ll normed spaces, the nuclear spaces which commonly appear in distribution theory, and also each conjugate Banach space with the bounded weak-star topology. It is shown that the theory of Banach- space d ifferential calculus can be extended with only slight m odification to produce a theory of differentiable maps between D -spaces in which most of the basic results of Banach-space d ifferential calculus adm it generalizations.

The fundam ental r e su lts of the d iffe ren tia l ca lcu lu s of m aps between fin ite-d im en sio n al v ec to r sp a c e s a re :

(1) the c lo su re of C k m aps under com position , k e N(2) the in v e rse function theorem(3) the ex isten ce of Ck flow s fo r Ck v e c to r fie ld s(4) the ex isten ce , fo r each open su b se t U of IRn, of a sm ooth function f:

IRn -» IR such that f(x) f 0'#=^-x e U (which im p lie s the ex isten ce of sm ooth p artitio n s of unity).

The extension of d iffe ren tia l ca lcu lu s to Banach sp a c e s has been com pletely su c c e ss fu l, in that (1), (2), (3) a r e s t i l l true in the m ore gen eral settin g; and even (4) holds, at le a s t fo r se p arab le H ilbert sp a c e s and certa in other nice Banach sp a c e s (due to J . W ells, unpublished).

A theory of d iffe ren tia l c a lcu lu s fo r m aps between o b jects in a catego ry of sp a c e s p ro p erly l a r g e r than the ca teg o ry of Banach sp a c e s m ust, at le a s t , s a t i s fy (1) and reduce to the u su al th eory fo r m aps between Banach sp a c e s . The u su a l e lem en tary r e su lts such a s T a y lo r 's theorem should gen era lize .

However, the fundam ental pu rp ose of such a theory should be to provide a foundation fo r g e n era liza tio n s of (2) and (3) which a r e of u se in the study of d iffe ren tia l equations.

In th is p ap er, we sh a ll outline a gen era l theory of d iffe ren tia l ca lcu lu s, and define a catego ry of sp a c e s and m aps fo r which it i s e a sy to prove a gen era lizatio n of (1). The defin itions have the advantage of being intuitive and e a sy to work with, and the c a teg o ry of sp a c e s i s la rg e enough to include C" (D", IR) and the sp ace of d istrib u tio n s on Dn. G en eraliza tio n s of (2) and(3) to a c la s s of n o n -m etrizab le lo c a lly convex sp a c e s which i s im portant in global a n a ly s is a lso e x ist , but these w ill not be d isc u sse d h ere (see R e fs [ 1, 2 ]).

263

264 GRAFF

Many of the id e a s in th is a r t ic le had th e ir o rig in s in p ap er [ 5] by K ijow ski and Szczy rb a .

L e t V and Z be lo c a lly convex sp a c e s , U an open su b se t of V with0 e U, and h: U -► Z a m apping of se ts :

1. Definition, h is tangent to 0 at 0 if:(a) h(0) = 0.(b) fo r each sem i-n o rm X on Z, there e x is t s a sem i-n o rm u o n V such

that: fo r each e > 0, there e x is t s a neighbourhood Ue of 0 with Ue c U, such that y e Ue= > X (h(y)) й e ■ v (y).

2. R e m a rk s , (a) h tangent to 0 at 0 = ^ h is continuous at 0.(b) If V and Z a r e norm ed sp a c e s , the above definition is equivalent to

the u su al condition

U h(y) Il S II у II • a( Il у I with a(r) -» 0 a s r -* 0

3. D efinition. L e t U be an open su b se t of V, f: U -» Z a m apping of se ts ,and x e U , We say that f i s (Fréch et) d ifferen tiab le a t x if there e x is t sSL £ L(V , Z) such that r x i s tangent to 0 a t 0, w here r x i s the m ap defined by rx(y) = f(x + y) - f(x) - i(y ) .

R e ca ll that L(V , Z) i s a lo c a lly convex sp ace , and is given the topology of uniform convergence on bounded su b se ts of V.

4. D efinition, f : U -» Z i s C 1 if f i s d ifferen tiab le at each point in U, andif D f: U - L(V , Z) i s continuous, f i s C k, к > 1, if Df i s C k_1. f i s C°° if fi s Ck fo r a ll к e N.

5. R e m ark . Note that, if f e L(V, Z), then f i s C ".If Z i s a lo c a lly convex sp ace , and X a continuous sem i-n o rm on Z,

let Nx = { x e Z: X(x) = 0 } , so that i s a c lo sed su b sp ace of Z . Define Ъ\ to be the norm ed sp ace where underlying lin ear sp ace i s Z /N \, with the norm which X in duces on th is lin e a r sp ace . L e t px : Z -» Z x be the n atu ral pro jection .

6. L e m m a . L e t V and Z be lo c a lly convex sp a c e s , U an open su b se t of V, f : U - Z , a n d x e U . Then:

(a) f i s d ifferen tiab le at x > p \ ° f : U -» Zx i s d ifferen tiab le at x fo r each sem i-n o rm X on Z.

(b) f is C 1< > p \ ° f i s C 1 fo r each sem i-n o rm X on Z.The above lem m a, while tr iv ia l to v erify , has im portant im p lication s:

fo r it im p lie s that, to study d ifferen tiab le m aps between lo ca lly convex sp a c e s , it su ff ic e s to study the sp e c ia l c a se of d ifferen tiab le m aps with norm ed sp a c e s a s the ran ge sp a c e s . T h is i s p re c ise ly the d irection in which we sh a ll now p roceed .

L e t V be a lo c a lly convex sp ace , E a norm ed sp ace , and v a sem i-n o rm on V. The sem i-n o rm v in duces a subadditive function V: L(V , E) -* IR+ u{oo} which i s defined a s fo llow s: fo r each £ e L (V , E), we le t v'(Æ) = sup { ||ü(v) || : i/(v) è 1}, w here | | j é ( v ) || i s the norm of ü(v) in the norm ed sp ace E . Since i^ is subadditive and non-negative, V "a lm o st" m akes L(V , E) into a norm ed sp ace , excep t that som e elem en ts of L(V , E) m ight have "n o rm " equal to infinity. So we sim p ly throw th ese e lem ents away, and m ake the following definition:

IAEA-SM-11/33 265

7. Definition. L^fV , E) i s the norm ed lin e a r sp ace co n sistin g of those e lem en ts í e L(V , E) fo r which Щ£) < °o, with ~v a s the norm function.

8. R e m a rk s , (a) The lin e a r in clusion L „(V , E) -» L(V , E) i s continuous.(b) F o r each h e L(V , E), there e x is t s a sem i-n o rm ц = ц(h) such that

h e i y v , E).Note that the second p art of the above rem ark im p lie s that

L(V , E) = U L „(V , E), w here A i s the se t of continuous se m i-n o rm s on V. v e л

9. L em m a. L „(V , E) i s can on ically iso m o rph ic to L (V „, E ).P ro o f. E ach functional in Ij(4 v, E) obviously induces an elem ent of L „(V , E), and the topology on both sp a c e s i s induced by a norm : the topology of uniform convergence on the unit у-b a ll of V. So to p rove the lem m a, it su ffic e s to show that each elem ent of L „(V , E) induces an elem ent of L(Vi/, E ). L e t £ e L „(V , E) and suppose x i s an e lem ent of V such that v(x) = 0. If we show that i(x ) = 0, then it follow s that SL co rre sp o n d s to an e lem ent of L(Vi/, E).Now, sin ce i e L „(V , E ), there e x is t s n e N such that j| jC (v ) || S n fo r a l l v e V with v(v) S 1. But y(rx) = 0 fo r a l l r 6 IR, which im p lie s that r ||i(x ) || È n fo r a ll r e IR, which im p lie s that ||i(x )|| = 0. Thus i(x ) = 0.

It i s now p o ss ib le to give a tentative definition of a c la s s of lo c a lly convex sp a c e s in which the study of d ifferen tia l ca lcu lu s i s fa ir ly sim p le .

10. D efinition. A D j-sp a c e i s a lo c a lly convex sp ace V with the follow ing p roperty : if U i s a neighborhood of 0 in V, E a norm ed sp ace , andf : U -» L(V , E) a continuous m ap, then there e x is t s a neighbourhood W of 0 with f f £ U , and a sem i-n o rm v on V, such that f(W) ç L „ ( V , E) and such that the m ap f : W -» L „(V , E) is continuous.

T he obvious ex am p les of D j- sp a c e s a r e the norm ed sp a c e s . T here e x is t other, n o n -triv ia l exam p les of D i- sp a c e s , but we w ill d e fer d iscu ssio n of th ese until we have e stab lish ed a few e lem en tary re su lt s about d ifferen tiab le m ap s between D j- sp a c e s .

11. T h eo rem . L e t V and Y be D i- sp a c e s , E a norm ed sp ace , U open in V,W open in Y, f : U -* W a C 1 m ap, and g:W -> E a C ' m ap. Then g о f i s С 1 . P ro o f . Note that D (g o f) e x is t s fo r each x e U , and D (gof)(x) = Dg(f(x)) • Df(x). Thus we need m e re ly show that D (g o f) is continuous. T o show th is, it su ffic e s to show that, fo r each x e U, there e x is t s a neighbourhood A of x inU such that D (g o f) i s continuous on A. So le t x e U , B y the definition of a D j-sp a c e , there e x is t s a neighbourhood В of g(x) and a sem i-n o rm ц on Y such that Dg(B) с L M ( Y, E) and such that Dg : В -* L M ( Y, E) i s continuous. S im ila r ly , there e x is t s a neighbourhood A of x (we m ay a ssu m e A c f _1 (B)), and a sem i-n o rm v on V, such that Dípô f)(A) с L U(V, Y and such that D (p jjo f): A -» L ¡,(V , Yjj) = L (V „,Y (j) i s continuous. Then, on A, we have the follow ing fac to riza tio n of D (g o f) ;

A D(pM° f ) -— g ) ° f - L (V y, Yjj) x L (Y „, E) - L (V „, E) = L „(V , E)

- L(V , E).

The b ilin e ar m ap L (V „, YM ) x L (Y M, E) -» L(V v , E) i s C“ sin ce V„ , У ц and E a re a ll norm ed sp a c e s , and the in clu sion L v(V, E) -» L(V , E) i s sm ooth, sin ce

2 6 6 GRAFF

it i s lin e a r . Since D(pMo f) : A -> L (V „, Y¡¡) and (D g)of : A -» Ь(УД, E) a re both continuous, we conclude that D (gof) i s continuous on A.

12. T h eo rem . L e t V and Y be D ^ -spaces, Z a lo ca lly convex sp ace , U open in V, W open in Y, f : U -> W a C 1 m ap, and g: W - Z a C 1 m ap. Then g o f i s C 1 .P ro o f. By p art (b) of L em m a 6, it su ffice s to show that p \o (g o f) is C 1 fo r each sem i-n o rm X on Z . But p x °(g o f) = (p xo g )o f. Now, sin ce g i s C 1, L em m a 6 im p lie s that p \o g i s C 1. Thus (p \o g )o f i s C 1 by the p rev iou s theorem .

One other re su lt which we sh a ll sta te is a co n v erse to the Fundam ental T h eorem of C alcu lu s, which i s e s se n t ia l to the p roof of T h eorem 19.

13. L e m m a . L e t V be a D i-sp ace , Z a com plete lo c a lly convex sp ace , U a convex open su b se t of V, and f : U -» Z . A ssu m e there e x is t s a continuous m ap g : U -» L(V , Z) such that

f(y) - f(x) = J g(ty + (1 - t)x) (y - x) dt fo r each x, y e Uо

Then f i s C 1, and Df = g.The e s se n t ia l id e a s of th is theory of d iffe ren tia l ca lcu lu s a re contained

in the C 1 theory d isc u sse d above. With th is theory a s m otivation, we now give the defin itions which we need to develop the C k theory.

L e t V be a lo c a lly convex sp ace , E a norm ed sp ace , r e N, and v a s e m inorm on V. Then v induces a subadditive m ap 7 : L C(V, E) -» IR+ c{oo} by

'v(i) = sup SL (v j, . . . , vr ) (I : v (v¡ ) S 1, 1 s i S r

14. D efinition. LÍ,(V, E) i s the norm ed lin e a r sp ace co n sistin g of thoseelem en ts i e L r(V, E) fo r which ’v(l) < oo, with V a s the norm function.

15. D efinition. A D -sp a ce V i s a lo ca lly convex sp ace with the follow ing property : if U i s a neighbourhood of 0 in V, E a norm ed sp ace , andf : U -* L r(V, E) a continuous m ap, then there e x is t s a neighbourhood W of0 with W c u , and a sem i-n o rm v on V, such that f(W) ç L j l V , E) and such that the m ap f : W -* L r¡,(V, E) i s continuous.

One im m ediate consequence of the above definition i s the following:

16. L em m a. If V is a D -sp a ce , Z a lo c a lly convex sp ace , and r e N, then L(V , L r (V, Z)) = L r+1 (V, Z).

17. R e m ark . The above lem m a im p lie s that L (V, L(V , . . . , L(V , Z)). . . . ) = L r(V, Z ). T hus, if f : V -* Z i s a C k m ap, then D ¿f i s a m ap fro m V to L ‘ (V, Z) fo r each 1 â i s k.

18. T h eo rem . L e t V be a D -sp a ce , Z a lo ca lly convex sp ace , U an open su b se t of V, f : U -* Z , and к e N. Then f i s C k"£=^p\of i s C k fo r each s e m inorm X on Z .P ro o f . U sin g R em ark 17, the p roof i s analogous to the p roof of L em m a 6.

IAEA- SM-11/33 267

The e sse n t ia l d ifferen ce between the theory of d ifferen tia l ca lcu lu s in the catego ry of D -sp a c e s and the theory in the ca teg o ry of D ]- sp a c e s i s the follow ing theorem , which follow s im m ediate ly fro m L em m a 13, R em ark 17, and the definition of a D -sp a ce :

19. T h eorem . L e t V be a D -sp ace , E a norm ed sp ace , U a neighbourhood of 0 in V, r and к non-negative in te g e rs , and f : U -» L r(V, E) a C k m ap. Then there e x is t s a neighbourhood W of 0 with W £U , and a sem i-n o rm v on V, such that f(W) Ç L r„(V, E) and such that the m ap f : W -» L^(V , E) i s C k.

20. T h eo rem . L e t V and Y be D - sp a c e s , E a norm ed sp ace , U open in V,W open in Y, and к e N. A ssu m e f : U -» W and g : W -* E a re Ck m ap s. Then g o f is C k.P ro o f . B y induction: we know the re su lt fo r к = 1. So le t к > 1, and a ssu m e the re su lt proved fo r the c a se (к - 1). U sing T h eorem 19, and the fa c to r iz a tion of D (gof) u sed in the p roof of T h eorem 11, it fo llow s that D (gof) is Ck_1, and hence that g o f is C k.

21. T h eo rem . L e t V and Y be D - sp a c e s , Z a lo ca lly convex sp ace , U open in V, W open in Y, and к e N . A ssu m e f : U -> W and g : W -* Z a re Ck m ap s. Then g o f i s C k.P ro o f . Im m ediate fro m T h eo rem s 18 and 20.

It i s now p o ss ib le to prove other b a s ic r e su lt s about d ifferen tiab le m ap s. F o r in stan ce , if V and Z a r e lo ca lly convex sp a c e s , U an open su b se t of V, f : U - L r(V, V), and g : U - L(V , Z), define the m ap g - f : U - L r(V, Z) by (g- f)(x)(w) = g(x)(f(x)(w)) fo r a ll w e Vr, x e U.

22. T h eo rem . L e t V be a D -sp a ce . A ssu m e that f : U -» L r (V, V) and g : U -» L (V, Z) a re both C k m ap s. Then g • f i s С k.

The above theorem i s u sefu l in the study of in v e rse function theorem s in the catego ry of D - sp a c e s . H owever, we sh a ll devote the rem ain d er of th is a r t ic le to showing that m any of the non-norm able function sp a c e s com m only encountered in a n a ly s is a re D -sp a c e s .

23. Definition. An exponential sp ace V i s a lo c a lly convex sp ace such that Vr i s com pactly gen erated fo r a ll r e N.

24. R em ark . Note that a m etrizab le lo c a lly convex sp ace i s an exponential sp ace .

L e t Z i and Z 2 be lo c a lly convex sp a c e s . F o r each r e N we defineev: L r(Z i, Z 2) x Z i -> Z 2 to be the evaluation m ap, i. e. ev(f, w) = f(w) fo r a llf e L ' ( Z 1 , Z 2), w e Z 'j , The follow ing lem m a is the fundam ental re su lt about exponential sp a c e s :

25. L em m a. L e t V be an exponential sp ace , Z a lo ca lly convex sp ace , U an open su b se t of V, r e N , and f : U - L r(V, Z) a continuous m ap. Then e v o (f x I): U x Vr -* Z i s continuous.

26. D efinition. A Schw artz sp ace V is a lo c a lly convex sp ace such that, fo reach sem i-n o rm X on V, there e x is t s a sem i-n o rm у on V such that v ê Xand such that the induced m ap i s p recom p act.

2 6 8 GRAFF

The definition of a Schw artz sp ace i s due to G rothendieck, and h is o rig in a l a r t ic le [3 ] rem ain s one of the b e st re fe re n c e s on the su b ject (another v e ry read ab le re fe ren ce is Ref. [4 ] ) .

O ur la s t re su lt w ill be that an exponential Schw artz sp ace i s a D -sp a ce . B e fo re giving the proof, however, we sh a ll l i s t som e exam p les of exponential Schw artz sp a c e s , and give som e definitions.

2 7. E x am p le s , (a) C “ (M, IR), the sp ace of sm ooth functions on any m anifold (com pact or non-com pact, with or without boundary).

(b) the sp ace of testin g functions on any open su b se t of IRn.(c) H'~(M , IR), the sp ace of d istrib u tion s on any com pact m anifold M.(d) 1R “ .(e) L e t В be a Banach sp ace . Then B ' with the bounded w e ak -sta r

topology i s an exponential Schw artz sp ace (d ifferen tia l ca lcu lu s in th is c a te gory of sp a c e s i s studied ex ten siv e ly in R e fs [1 , 2]).

2 8. D efinition. L e t Z i and Z 2 be lo c a lly convex sp a c e s , and le t r e N.(a) L s (Z j, Z 2) i s the l in e a r sp ace of r - l in e a r m aps from Z i to Z 2 with

the topology of pointw ise convergence.(b) L^Z -l, Z 2 ) i s the l in e a r sp ace of r - l in e a r m aps from Z j to Z 2 with

the topology of uniform convergence on p reco m p act su b se ts of Z i .The above function sp a c e s w ill be u sed in the p roof that an exponential

Schw artz sp ace i s a D -sp a ce . We sh a ll only be in te re sted in the c a se when Z x and Z 2 a r e norm ed sp a c e s . In th is c a se we have the follow ing lem m a, which i s an im m ediate consequence of the definition of a p recom pact set:

29. L em m a. L e t E i and E 2 be norm ed sp a c e s , let r e N, and a ssu m e that A i s a bounded su b se t of L r(E i, E 2 ). Then the su b sp ace topology induced onA by L jiE x , E 2) co in cides with the su b sp ace topology induced on A by L c (E 1; E 2).

30. T h eo rem . L e t V be an exponential Schw artz sp ace . Then V is a D -sp a ce . P ro o f. L e t E be a norm ed sp ace , U a neighbourhood of 0 in V, r e N ,and f : U -* L r (V, E) a continuous m ap. We m ust find a neighbourhood W of 0and a sem i-n o rm v on V such that f(W) с L£(V, E) and such that the m apf : W -> L^(V , E) i s continuous. By L em m a 25, the induced mapg = e v o (f x I) : U x V 1 -» E i s continuous. Since g i s continuous, and g(0, 0) = 0,there e x is t s a neighbourhood W of 0 and a sem i-n o rm X on V such thatIl g(x, w) II S i fo r a l l x e W, and fo r a ll w = ( v j , , , , , vr ) e V r such thatX(v¡) S 1 fo r a ll 1 S i S r .

T hus, f ( W )c L \ (V ,E ) = L r (Vx, E).Unfortunately, f m ight not m ap W continuously into L r(Vx, E ). However, sin ce g i s continuous, the m ap gw: W -* E which i s defined by

W-^— L r(Vx , E ) - ^ E

x ---- - f (x )----- f(x)(w)

i s continuous fo r each w e Vx, which im p lie s that the m ap f : W -» LjCV^, E) i s continuous. Note that'Tfffy)) S 1 fo r a ll x eW , so that the topology on f(W) a s a su b sp ace of LljCVx, E) i s the sam e a s the topology which f(W) in h erits a s a su b sp ace of L ç (V \, E ). Thus f : W -> L ç (V \, E) i s continuous.

IAEA-SMR-11/33 2 6 9

F in a lly , choose a sem i-n o rm v o n V, i / U , such that the induced m ap V v -> V \ i s p recom p act, which im p lie s that the induced m ap L c (V \, E) -* Ь Г(У„, E) = L r„(V, E) i s continuous. Then f(W) с L r„(V, E), and f : W -> L lv(4, E) is continuous.

T h is concludes our d isc u ssio n of e lem en tary d iffe ren tia l ca lcu lu s. F o r a m ore com plete treatm ent, and ap p lication s to global a n a ly s is , see R e fs [ 1, 2 ] .

R E F E R E N C E S

[1] GRAFF, R ., Elements of local non-linear functional analysis, Ph. D. Thesis, Princeton University, Princeton, N.J- (1972).

[2] GRAFF, R ., Elements of non-linear functional analysis, in preparation.[3] GROTHENDIECK, A . , "Sur les espaces (F) e t (DF)", Summa Brasiliensis 3 (1954) 57.[4] HORVATH, J . , Topological Vector Spaces and Distributions, Addison-W esley, Reading, Mass. (1966).[5] KIJOWSKI, J . , SZCZYRBA, W ., On differentiability in an im portant class of locally convex spaces,

Studia Math. 30 (1968) 247.

IAEA-SMR-11/34

SINGULARITIES IN "SOAP BUBBLES" AND "SOAP FILMS"

Jean E. TAYLOR Departm ent of M athem atics,Rutgers University,New Brunswick, N. J . ,United States of Am erica

Abstract

SINGULARITIES IN "SOAP BUBBLES" AND "SOAP FILMS''.The geometry of some surfaces arising in the calculus of variations and physically represented by soap

bubbles and soap films is discussed.

In th is p ap er, we d is c u s s the geom etry o f som e su r fa c e s which a r i s e in the ca lcu lu s of v a r ia t io n s , su r fa c e s such a s a re rep re sen ted ph y sically by soap bubbles and soap f i lm s . T here h as been an ex ten sive lite ra tu re on the stru c tu re of soap bubbles and film s, notably including P la te a u 's two vo lum es on the su b ject [ 1 ] ; how ever, it i s only recen tly that the su r fa c e s which m u st be co n sid ered in the ap p ro p ria te m inim izing p rob lem have been given a thoroughly r igo ro u s m ath em atica l form ulation .

B e fo re con sid erin g the m ath em atica l m odel, le t us look at the e x p e r im en tal o b se rv atio n s of ac tu al soap bubb les. In F ig . 1 there i s a sketch of th ree soap bubbles stick ing togeth er. If one looks at such a compound bubble, one notes that it c o n s is ts o f se v e ra l (s ix h ere) "sm ooth su r fa c e s " each having constant m ean cu rv atu re , togeth er with se v e r a l (four here) "sm ooth a r c s " along which three of th ese su r fa c e s m eet at 120 ° an g le s — that i s , along these a r c s the su r fa c e looks tan gentia lly like F ig . 2 — and se v e ra l (two here) iso la te d "p o in ts" at which four such sin g u la r a r c s m eet, bringing together s ix sm ooth su r fa c e s , a s in F ig . 3. That th ese a re the only phenom ena which occu r stab ly (that i s , which p e r s is t under sm a ll p ertu rb ation s such a s shaking and blowing) in any soap film o r bubble i s p re c ise ly P la te a u 's ob servation .

A reaso n ab le m ath em atical form ulation o f the p rob lem which the p h y sic s o f soap bubb les se e m s to im pose i s that o f en closin g constant vo lu m es with the le a s t p o ss ib le to ta l su r fa c e a re a , a s h as been recogn ized fo r m any y e a r s . P r e c is e ly , le t a j , . . . , a N be n um bers g r e a te r than ze ro , and let jd be the co llection o f a ll s e t s of N open s e t s (Aj, . . ., A N) with vo lum e1 (A¡) = a ¡ fo r each i = 1, . . . , N. The p rob lem i s then to find (A1, . . . , A' ) e js ' such that

NA r e a ( и boundary A!)

i = i 1

1 Throughout this paper "volum e", "area" , and "length" mean precisely Hausdorff 3 -, 2 - , and 1-dim ensional measure, respectively. Hausdorff measure agrees with any other reasonable definition of area (respectively, volum e or length) on smooth submanifolds, but additionally gives a precise meaning to area when singularities m ay be present.

271

272 TAYLOR

i s a m inim um o v er a ll jtf. If (A^, . . . , A^) g iv e s such a m inim um , then

N(boundary A.1)

w ill be ca lled a "so a p bubb le".S im ila r ly , given a com pact se t В o f finite length, we define the c lo sed

se t S to be a "so a p film " with boundary contained in В if the a re a o f S i s l e s s than or equal to the a r e a o f any su r fac e S' obtained from S by a C" deform ation of R3 which does not m ove B .

E a ch of these fo rm u latio n s im p lie s a v a r ia tio n a l fo rm u la . F o r a " so a p bubble" S, it i s that

A rea (S П B(p , r)) é (1 + K r) A rea (f(S n B(p, r)))

fo r a ll p in S, a ll r l e s s than som e constant 6 l a r g e r than 0, and a ll deform ation s f such that {x: f(x) j- x } U {y : y = f(x) f x} i s contained in B (p , r), w here B (p , r) i s the c lo sed (th ree-d im en sion al) b a ll centred at p o f ra d iu s r and К i s a constant (which tu rn s out to be dependent on the m axim um m ean cu rvatu re of the analytic p art o f S). F o r a "so a p film ", the correspon din g inequality i s the sam e except that we req u ire К = 0 and r < d ist(p , B ).

It h as recen tly been shown that solu tions to th ese p ro b lem s do alw ays e x is t and that the re su ltin g "so a p f i lm s" and "so a p bubb les" a re , in fact, except fo r a com pact sin gu lar se t of zero tw o-dim ensional a re a , the union of tw o-d im ensional an alytic subm anifolds of R3 [ 2 ] . We co n sid er here the rem ain ing se t, the sin gu la r se t . Sp ecifica lly , we sh all c la s s ify a ll the p o ss ib le s in g u la r it ie s , g ive the lo c a l com bin atoria l and d iffe ren tia l stru c tu re of the su r fa c e s n ear the s in g u la r it ie s , and show that th is c la ss if ic a t io n ,

FIG. 1. A compound bubble.

IAEA-SMR-11/34 273

FIG. 3. The cone T , re s tr ic ted to the convex h u ll o f its ve rtices .

in fac t, co in c id es with P la te a u 's ax io m s. (The w ork outlined in th is paper is su b stan tia lly contained in the au th o r 's d octora l d isse r ta t io n [3 ] and w ill ap p ear in d etail la te r [ 4 ] . )

A nice p ro p erty of a r e a p ro b lem s in the ca lcu lu s of v a r ia t io n s, a s tre a te d in the context of geo m etric m e a su re theory, i s that tangent cones e x is t at a ll points in so lu tions to the prob lem . A su rfac e С is defined tobe a tangent cone to a su rfac e S at a point p if there e x is t s a sequence ofra d ii r ; -*• 0 such that

С = lim £ ( l / r ¡ ) т(р) (S П B(p, r ¡ )) i-> 00

where

т(р) : R3 - R 3 , т(р)(х) = x - p

£ ( l / r ¡ ) : R3 - R3 £ ( l / r . )(x) = x / r t

B(p , r) = R3 n íx: I x - p I s r } a s b efo re .

274 TAYLOR

FIG.4. A spiral having every ray as a tangent cone.

T here m ay e a s i ly be m ore than one tangent cone at a point; fo r in stan ce, co n sid er the 1-d im en sion al sp ir a l of F ig . 4. C onceivab ly, there m ight a lso not be any tangent con es at som e point. H ow ever, if the su rfac e s a t i s f ie s the v a r ia t io n a l inequality above, then one can show that fo r sm a ll r

r 2 e 1 . A rea (T n B(p, r))

d e c r e a se s m onotonically a s r d e c r e a se s to z e ro . One th erefo re h as an upper bound to the a r e a s o f the su r fa c e s

g[l/r) r[p) (T П B(p, r))

fo r sm a ll r ; sin ce the c la s s e s o f su r fa c e s one usually co n sid e rs in geo m etric m e asu re theory have stron g co m p actn ess p ro p e rtie s in the weak topology, fo r any sequence o f ra d ii d ec rea sin g to zero som e subsequence o f the c o r r e sponding sequence of su r fa c e s m u st converge.

T h ese tangent cones can be shown to have se v e ra l p ro p e rtie s :(1) They a re con es ( i .e . they c o n sis t of r a y s from th eir boun daries to the orig in )(2) They a re a r e a m inim izing (that is , A rea(C ) § A rea (f(C)) fo r any deform ation f: R3 -» R? which does not m ove ЭС).

T h ere fo re , to a sk what tangent cones a re p o ss ib le i s to a sk "w hat a re a ll the area -m in im iz in g tw o-dim ensional cones in R 3? " . Since we are con sid erin g con es, it i s su ffic ien t ju st to ch arac te r iz e th e ir b ou n daries.Two conditions on th ese a r i s e a lm o st im m ediately : ( l) 'th a t the boundaries c o n sis t o f segm en ts o f g re a t c ir c le s , and (2)' that th ese segm en ts in te rse c t

IAEA-SMR-11/34 275

only 3 at a tim e at 120° an g le s . We note that, by the G aass-B on .net theorem , which g iv e s the a re a of a face bounded by g e o d e s ic s a s 2ir- L (ex te r io r an g les at v e r t ic e s ) , the m axim um num ber o f v e r t ic e s fo r any face outlined by g e o d e s ic s a s in (2)' above i s five . The problem of finding a ll con figuration s sa tisfy in g (1 )' and (2)' above should th erefo re be so lub le; and in fac t the only p o s s ib il it ie s are a s follow s:

(1) A sin gle g re a t c irc le ;(2) 3 half c ir c le s at 120° an g les (the cone o v er th is boundary is

p re c ise ly F ig . 2),(3) 6 segm en ts o f g re a t c ir c le s form ing the on e-skele ton o f a re g u la r

sp h e rica l tetrah edron (the cone over th is boundary i s a s shown in F ig .3 ) ,(4) 12 segm en ts form ing the one skeleton o f a sp h e r ic a l cube,(5) 9 segm en ts form ing the on e-skeleton of a sp h e r ic a l p r ism over

a re g u la r tr ian g le (only one such figure e x is t s sa tisfy in g (1 )' and (2)' above),(6) 15 segm en ts form ing the on e-ske le ton o f a sp h e r ic a l p r ism o v er

a re g u la r pentagon (again, there i s only one of th ese),(7) 3 0 segm en ts form ing the on e-skeleton o f a re g u la r sp h e rica l

dodecahedron,(8) 24 segm en ts form in g 2 q u a d r ila te ra ls and 8 pentagons,(9) 21 segm en ts form in g 3 q u a d r ila te ra ls and 6 pentagons,

(10) 18 segm en ts form in g 4 q u a d r ila te ra ls and 4 pentagons.We now a sk which o f the above con figu ration s actu ally sa t is fy the

p rop erty that th e ir con es a re a r e a m in im izing; we see that (1), (2), and (3) do, w h ereas we can co n stru ct ex p lic it d eform ation s which sav e a re a com p ared to the cones over (4), (5), (6), (7), and (10).

To il lu stra te how the above i s proved , we show here that the only configuration where a ll fa c e s have four v e r t ic e s i s the regu lar-on e (that is , where a ll ed ge s have equal length), and that the cone ov er th is configuration is not a rea -m in im iz in g .

FIG. 5. A spherical quadrilateral, extended.

276 TAYLOR

A ssu m e Q i s a sp h e r ic a l q u a d r ila te ra l and one side h as length a . T ake the two ad jacen t s id e s to a, and extend each of them (a s g e o d e sic s ) in both d irectio n s until they in te rse c t (see F ig . 5). Now e ith er two an g les and the included sid e o r 3 an g le s com pletely determ ine a sp h e rica l tr ian g le ; th e re fore given the one sid e length a, the angle o f in tersec tio n of the extended g e o d e s ic s i s determ ined , the other angle o f in te rsec tio n m u st be the sam e, and th ere fo re the sid e of the q u a d r ila te ra l opposite to the side of length a m u st be o f length a a lso . We fu rth er se e that the two other s id e s m ust be o f equal length, and that that length i s determ ined by a . T h ere fo re there i s only a one d im en sion al fam ily of q u a d r ila te ra ls on the sph ere (with 120° an g le s), and they a re a ll re c ta n g le s .

Now co n sid er th ree ad jacen t such q u a d r ila te ra ls , a s in F ig . 6 . If one of them h as sid e lengths a and ¡3, then the th ird edge 7 in tersectin g these two m u st have length a from con sideration of the second q u ad rila te ra l and /3 from con sid eration of the th ird . T h ere fo re , a = ¡3 = 7 , and a ll s id e s are equal ( i .e . the configuration is that of the on e-skeleton of a sp h e rica l cube).

The cone ov er the one skeleton of a cube i s not area-m in im izin g , a s can be seen by dipping a w ire fram e form ed into such a boundary in a soap solution . The re su lt w ill be a s d iagram m ed in F ig . 7; such a su rfac e can in fac t be obtained from a deform ation of R3 applied to the cone. H ow ever, the s im p le r deform ation which ju st m ak es the cen tra l su r fac e a sm a ll sq u are o f s id e d can be shown it s e lf to have a r e a at le a s t 0. 06d2 l e s s than the a r e a o f the cone; th ere fo re the cone i s not area-m in im izin g .

The re su lt s sta ted above can be com bined into the following th eorem .T h eorem 1 [ 4 ] . The only tangent cones to "so a p b u bb les" and

"so a p f i lm s" a re the d isk , the cone Y o f F ig . 2, and the cone T o f F ig .3 .Having th is th eorem , we have a g ra sp on the lo c a l s tru c tu re o f soap

bubbles and film s; how ever, a s yet, we have given no indication whether behaviour of the type of F ig . 4 can o ccu r. H owever, we can prove m uch m ore .

IAEA-SMR-11/34 277

T h eorem 2 [ 4 ] . L e t S be a " so a p f ilm " o r " so a p bubb le". Then(1) T h ere a re unique tangent con es at ev ery point,(2) The points with Y a s a tangent cone form a 1 -d im ensional H ôlder-

continuously d ifferen tiab le subm anifold of R3 .(3) In a neighbourhood of a point with Y a s a tangent cone, S c o n sis ts

of three tw o-d im ensional H older-continuously d ifferen tiab le m an ifo ld s- w ith-boundary m eeting at 120° an g le s along the subm anifold of ( 2); in a neighbourhood of a point with tangent cone T, S c o n s is ts o f 6 H older- continuously d ifferen tiab le m an ifo lds with c o rn e r s m eeting tan gentially a s in F i g . 3.

The m ain an aly tic e stim ate of the p ro o f i s an inequality re la tin g the a r e a of a su r fa c e within a ba ll to the length o f i t s boundary on the boundary o f the b a ll. The p a r t ic u la r inequality proven is analogous to one studied by the late E .R . R eifen berg and term ed by him an ep ip e rim e tr ic inequality . R eifen berg proved such an inequality when the tangent cones w ere assu m ed to be d isk s and the su r fa c e a ssu m ed to be area-m in im izin g (in gen era l d im en sio n s and cod im en sion s, how ever) [ 5 ] ; we p rove h ere such an inequality fo r sin g u la r tangent con es (of d im ension 2 in R 3 ) to m ore gen era l su r fa c e s by a m ethod to tally d ifferen t from R e ife n b e rg 's .

L e t us f i r s t define a p a ra m e te r which m on itors the behaviour of the su r fa c e at sm a ll ra d ii. We a lread y have that

r "2 e Kr A rea (S П B(p, r))

d e c r e a se s m onotonically to a lim it a s r ap p roach es ze ro ; th is lim it i s e a s i ly seen to be the a r e a o f a tangent cone at p. F o r s im p lic ity , le t ush ere only co n sid er poin ts with Y a s a tangent cone; th is num ber i s then3v/2. We th ere fo re define, at such points p,

E x c B (S, p, r) = r "2 e Kr • A rea (S П B(p , r)) - 3ir/2

Note that th is quantity i s a lw ays non-negative, by the m onotonicity property stated above.

278 TAYLOR

U sin g the notation p Ж W to denote the cone from a point p to a boundary W, we m ay now e x p r e s s the ep ip e rim e tr ic inequality we w ish to prove a s follow s:

T h ere e x is t к > 0, 6 > 0, e > 0, and J? < oo such that if

0 < E x c B (p X 8 (S П B(p, r)), p, r) < eand

0 < d istan ce ( g ( l / r ) T p ( p X 3 ( S n B ( p ,r ) ) ) , 0Y) < 6

fo r som e SO(3), the group of orthogonal ro tatio n s o f R 3 , then there e x is t s a su rface ' Z obtained from S n B(p, r) by a deform ation of R 3 and with 9Z = Э (S П B(p, r)), sa tisfy in g

E x c B (Z, p, r) S (1 - k) E x c B (p Ж 9 (S П B(p, r))) + ÍK r

In other w ords, except fo r the term iK r , one can sav e a r e a p roportion al to E x c B in the cone p X 9 (S П В (p, r )).

If such an inequality can be proved , then by the ex isten ce of tangent con es the conditions a re m et fo r sm a ll enough ra d ii (and, in fac t, they a re m et uniform ly fo r a ll p ' with tangent cone Y in a neighbourhood o f p).Since S s a t i s f ie s the v a r ia t io n a l fo rm u la given b efo re , we have fo r sm a ll ra d ii

E x c B (S, p, r) = r ' 2 e 1 1 A rea(S n B(p, r)) - 37r/2 ê e2Kr A rea(Z n B(p, r ) ) r ’ 2 - 3тг/2

á (1 - к) [ A rea(p X 9 ( S n B ( p , r ) ) ) e Kr r "2 - 3 7r / 2 ] +Î. K r + 2K r • A rea(Z П B(p, r ) )r 2

S (1 - k) [ (r /2 ) [(d/dr) A rea(S П B (p , г))] еКг r ’ 2 - Зтг/2] + £ 'K r [7, 4 .2 . 1]

â (1 - к) [ (r /2 ) ( d /d r )E x c B(S, p, r) + E x c B(S, p, r )] + Í 'K r

T h is inequality can now be in tegrated in the ap p rop ria te reg io n s to give

(Зтг/2)г2 s e Kr • A rea(S П B (p , r)) s (Зтг/2)г2 (1 + A 1 r 2k/(1 ' k> )

where A x i s a constant which can e a s i ly be ca lcu lated . T h is upper bound to the grow th in a r e a o f S n ear p l im its the am ount o f tw isting that can take p lace ; one can in fac t show that such a condition on the growth in a r e a ofa su r fa c e im p lie s unique tangent cones and a ll the other con clu sion s of theth eo rem . (R eifenberg obtained con clu sion s analogous to (1) and (2) from h is ep ip e rim e tr ic inequality but through a d ifferen t p ro o f [ 6 ] . )

The p ro o f o f such an ep ip e rim e tr ic inequality i s thus the h eart of the d ifficu lty . It i s done by contradiction ; th ere fo re th ere a re no a p r io r i e s t im a te s on k, the H older exponent. The contradiction i s a s follow s:

F o r ev ery v - 1, 2, . . . there e x is t n um bers k„ > 0, > 0 , > 0,lv < oo, and ru > 0, and su r fa c e s S„ with

(1 ) lim „k „ = 1н п „е„ = lim^ó,, = 0, 11т„4„ = oo

(2) E x c B(Sy , p , r J = ev

IAEA-SMR-11/34 279

(3) d is t ML/r)Tjp)(p X 3(S„ П В ( р , г Д Y) = 6V, and

(4) E x c B (Z, p, r^) > (1 - k) e„ + fo r a ll Z with 3Z = 3(S„ n B(p, r „ ) ) .

We m ay a lso a ssu m e

(5) If l i m ^ e ;1 = 0, then l im ^ ^ r ^ e ÿ 1 = 0

We m ay, in fac t, a ssu m e

(6) l i m ^ e " 1 = 0

sin ce o therw ise we im m ediate ly con trad ict the contradiction .The p ro o f that the con trad iction cannot be true i s ra th er technical;

one f i r s t su cce e d s in reducing it to the c a se where 3 (S ,r iB (p , rv) c o n s is ts of a fin ite num ber o f C1 a r c s fo r a ll la rg e v, and then to the c a se w here it, in fac t, c o n s is ts o f only three a r c s . Both of these argu m en ts involve showing one can push (by a deform ation of R 3) the given cone onto the cone o v er such a boundary, p rovided one adds a "p atch " out at the boundary; in o rd e r fo r th ere not to be an im m ediate contradiction , the a r e a of the "p atch " m u st i t s e lf be in sign ifican t in the lim it and can be ignored .

Once one i s dealing ju s t with seq u en ces o f con es o v er 3 a r c s , one d efin es a new e x c e s s E xcyfS^) a s fo llow s. The cone ov er the three a r c s , when tran sla te d so that it s cen tre i s at the o rig in and expanded to unit ra d iu s , m u st lie c lo se to som e orthogonal rotation of Y, by the d istan ce hypothesis. One ch o o se s the "b e s t p o s s ib le " such rotation and extends rad ia lly the cone o v er each sin g le a rc in turn until the boundary o f the extension p ro je c ts orthogonally , by the p ro jectio n defined by the ap p ro p ria te half plane o f Y, into the unit c irc le in that plane cen tered at 0. One then defines

3

E x c Y(S y) = ^ [(A reao f exten sion o f cone o v er ith a rc ) —

i = 1 (A rea of p ro jec tio n o f th is extension)]

T h is new e x c e s s i s proven to be re la ted to the old in the following two w ays:

lim „ E x c Y(S„) = 0 (easy)

E x c B(Si;, P. r v) á E x C y iS J (hard)

Noting that the extended su r fa c e can be reg ard ed a s lying within the o rig in a l cone by sim p ly shrinking it by a fac to r of 1 / 2 , one concludes that if the contrad iction o f the e p ip e rim e tr ic inequality holds fo r E x c B, it a lso holds (with d ifferen t seq u en ces k * , ó * , e * ) fo r E x c y .

But now we have su r fa c e s lying o v er th e ir p ro jec tio n s, when we co n sid er each a rc independently, and we can apply the pow erful m ethods o f A lm gren[ 2 ] to conclude that the su r fa c e s m u st, in fac t, be v e ry c lo se ( i .e . within a constant t im e s ExC ytS^)1 2) to being the g rap h s o f harm onic functions.The two conditions of being a cone and being v ery c lo se to harm onic im ply that the su r fa c e i s v e ry c lo se to a p lane; the three p la n es corresp on din g

2 80 TAYLOR

to the th ree a r c s cannot a ll be at 120° to each other b ecau se the E x c Y h as to com e from som ew h ere. One show s that, at le a s t , one o f the an g le s between the p lan es m u st d iffe r from 120° by at le a s t a constan t t im e s E x c y ^ , ) 1/2 .It i s then p urely a geo m etric argum ent to see that if the an g les d iffe r from 120° by с • (E x c y(Sb ))х 2 , then one can deform the p lan es n ear the centre line until they a re at 120° an g le s and in so doing sav e a re a p roportion al to E xcy (S^ ). But th is g iv e s us a final contrad iction of the contradiction , and the e p ip e rim e tr ic inequality m u st th ere fo re hold fo r som e k > 0, 6 > 0, e > 0 and i < oo. /

The p ro o f of the correspon din g inequality fo r points with T a s a tangent cone i s en tire ly analogous except that one red u ce s the problem to the ca se o f s ix a r c s ra th er than th ree .

A s we have seen a lread y , such an inequality is su ffic ien t to prove T h eorem 2. In fac t, if an analogous inequality could be proved fo r a r e a - m in im izing su r fa c e s o f h igher d im ension and codim ension , then one could ju st a s e a s i ly p rove the uniqueness of tangent con es and other strong conditions on the lo c a l com bin atoria l and d ifferen tia l stru c tu re of the su r fa c e s . So it i s a R eifen berg-type inequality, com parin g the a re a of a su r fa c e within a b a ll to that of the cone ov er it s boundary, that u n derlies the solution to the p rob lem , and it i s the tran sla tio n o f the p rob lem from one o f cones in b a lls to one o f s im p le r cones having nice p ro jec tio n s that con stitu tes the m a jo r p r o g r e s s involved in proving the re su lt outlined in th is p ap er.


T h is w ork w as supported in p art by a N ational Science Foundation G raduate F ellow sh ip .

R E F E R E N C E S

[ 1] PLATEAU, J. A .F ., Statique Expérim entale e t Théorique des Liquides Soumis aux Seules Forces M oléculaires, G authier-V illars, Paris (1873).

[2 ] ALMGREN, F .J . , J r ., Existence and regularity almost everywhere of solutions to e llip tic variational problems with constraints (in preparation).

13] TAYLOR, Jean, Regularity of the singular set of two-dim ensional area-m inim izing flat chainsmodulo 3 in R3, Ph.D. thesis, Princeton University (1973).

[4 ] TAYLOR, Jean, The structure of singularities in soap-bubble- and soap-film -type m inim al surfaces(in preparation).

[5 ] REIFENBERG, E .R ., An Epiperimetric Inequality related to the analyticity of m inim al surfaces, Ann. Math. 80 (1964) 1.

[6 ] REIFENBERG, E .R ., On the analyticity of m inim al surfaces, Ann. Math. 80^(1964) 15.[7] FEDERER, H ., Geometric Measure Theory, Springer-Verlag, New York (1969).

IAEA- SMR-11/35

CAUSTICS

J. GUCKENHEIMER Institute of Mathematics,University of Warwick,Coventry, Warks,United Kingdom

Abstract

CAUSTICS.After a description of light caustics as they arise in geom etric optics their relationship with Riemannian

geometry is described; the Hamilton*Jacobi theory is dealt with and the solutions of first-order partial differential equations are discussed. Finally, the connection of catastrophes and caustics is described.

T h is i s an ex p o sito ry p ap er on the s in g u la r it ie s of solu tions of f i r s t - o rd e r (non-linear) p artia l, d iffe ren tia l equations. The f i r s t sectio n of the p ap er is a d e scrip tio n of light c a u st ic s a s they a r i s e in geom etric o p tics . The re la tion sh ip of light c a u st ic s with R iem annian geom etry i s d e sc r ib ed . The second section d ea ls with the H am ilton -Jacob i theory and the solu tions o f f i r s t - o r d e r p a r t ia l d iffe ren tia l equations. The final sec tio n exp lo res the re la tion sh ip of s in g u la r itie s of so lu tion s of f i r s t o rd e r P D E 's with T h o m 's theory of c a ta stro p h es.

We do not docum ent the h isto ry o f th ese id e a s , but only re m ark that th e ir o rig in s a re deeply em bedded in o p tics, m ech an ics, and d ifferen tia l geom etry . The conceptual innovation of the p a st few y e a r s which allow s the su b ject to be trea ted sm oothly is the concept of a L agran gean m an ifo ld . T h is id ea se e m s f i r s t to ap p ear in the work of the R u ssian p h y sic ist M aslov and h as been studied subsequently by Arnold [1 ] , W einstein [14, 15 ], and H orm ander [5 ] . A b r ie f introduction to T h om 's theory of catastro p h es a p p e ars in R ef. [1 1 ] . A m ore detailed v e rs io n of our work in this d irection w ill ap p ear in R ef. [4 ]

1. LIGHT CAUSTICS

G eo m etric op tics i s a m ethod for studying approxim ate so lu tion s of M axw ell's equations. To em ph asize the geom etric nature of the theory, we d e sc r ib e it in the c o -o rd in a te -fre e context of a R iem annian m anifold M. (The index o f re fra c tiv ity of M d eterm in es it s R iem annian m e tr ic .) In the theory of geo m etric o p tic s , light c o n sis ts of p a r t ic le s which p rop agate along light r a y s . The paths of light ra y s a re g e o d e s ic s . They can be d e scrib ed from the in teg ra l cu rv es of a H am iltonian vecto r field defined on the cotangent bundle T *M of M. R e c a ll the p ro ced u re : The m etric on M induces an iso m o rp h ism of the tangent and cotangent bundles of M. U sin g this iso m o rp h ism , we re g a rd the m etric of M a s defining an inner product

281

282 GUCKENHEIMER

( , ) on each fib re of T *M . D efine the function H: T *M -* R by H (u) =I ш У . Then H is a sm ooth function which we take a s the H am iltonian function for defining a H am iltonian vecto r field X H on T *M .

The H am iltonian vecto r field X H is defined a s fo llow s: T h ere is a can on ical two form S7 on T *M . (See sectio n 2 for the definition of Í2.)П is c lo sed and of m ax im al rank; hence it defines a sy m p lectic stru c tu re on T *M . D efine the iso m o rph ism

£ ; T (T*M ) - T *(T *M ) by

£(Xi) (X2) = Q(XU X2)

for a ll X i, X 2 G Tp (T *M ), p € T *M . The m ap I is an iso m o rp h ism sin ce Q is a b ilin ear form of m axim al rank on each fib re of the tangent bundle of T *M . We now define X H = JT1 (dH).

The g e o d e sic s of M a re the p ro jec tio n s of in tegra l cu rves of X H. The in teg ra l cu rv es of X H lie on the lev e l su r fa c e s of H in T *M . R e str ic t in g our attention to a sp e c ific lev e l su r fa c e (say H = 1) co rre sp o n d s to fixing the speed of light. The u su a l d escrip tio n of the geod esic flow of M takes p lace in the tangent bundle ra th er than the cotangent bundle. By working in the cotangent bundle we can re in te rp re t the m eaning of the geod esic flow in te rm s of w avefronts and Huyghens' p rin cip le .

Up to a s c a la r fac to r , each non-zero covector at pG M is determ ined by it s k ern el which is a hyperplane in the tangent sp ace to M at p. Now think of a w avefront of light W propagating through M. W is a h y p ersu rface of M and it s tangent sp ace at each point i s a hyperplane in TM . T h ere is a line of co v ecto rs in T *M at each point of W having the tangent sp ace of W a s kern el. C hoosing the d irection of propagation of W and the speed of light g iv es u s a unique covecto r a sso c ia te d to W at each point. T h is covector is defined a s one o f the points of in te rsec tio n of the line of covecto rs annihilating the tangent sp ace of W with the sph ere w hose rad iu s is the speed of light. Thus there i s a m ap j : W -» T *M . If we follow j(W) along the geo d esic flow for tim e t and p ro jec t th is subm anifold of T#M into M, we obtain the new p osition Wt of the w avefront W a fte r it has m oved fo r t units of tim e . The iso m o rp h ism betw een T *M and TM induced by the R iem annian m e tr ic a s s o c ia te s to each non-zero covector a norm al to its kern el (of the sam e length and with the p ro p er orien ta tion ). In te rm s of a propagatin g w avefront of light this im p lie s a form of H uyghens1 p rin c ip le : a w avefront of light p ro p agate s n orm al to it se lf . F o r a treatm en t of catastrop h e theory from th is point of view , se e R ef. [9 ].

It i s p o ss ib le to so lv e for the su c c e s s iv e w avefronts Wt CM m ore d ire c tly than we did above by in troducing the H am ilto n -Jaco b i p a r t ia l d iffe ren tia l equation. In our c a se of geo m etric o p tics , we seek a sm ooth function f : M -* R such that \ <C df, df У = 1 where we have taken the speed of light to be 1. The function f should sa t i s fy the in itia l condition that f I W = 0 . In this c a se , each lev e l su r fa c e of f w ill be the se t of points at a given geo d esic d istan ce from W and w ill be one of the su r fa c e s Wt. In try in g to g lobally con struct functions f sa tis fy in g the above equations with given in itia l conditions, one frequently encounters d ifficu ltie s . T h ese ob stru ction s a re our cen tra l in te re st ; the goal i s to ach ieve a geom etric understanding of them .

IAEA-SMR-11/35 2 8 3

To il lu s tra te the nature of the prob lem , con sider an exam ple . T ake the p arab o la P = {(x , y ) | y = x2} a s in itia l w avefront W in the plane M with the d irectio n of propagation being upw ards. To ca lcu late the function f sa tis fy in g the p a r t ia l d iffe ren tia l equation

3f о 9f 2 f— Г + (— ) Эх 1Эу; 1

with f j P = 0 , con sid er the function g :R -* R defined by

g(x, y, s) = [ ( x - s )2 + (y - s 2)2

The v a r ia b le s p a ra m e tr iz e s the points of P , and g d eterm in es the d istan ce from the point (s , s 2) on P to the point (x, y). At a point (x, y) on the w avefront Wt at tim e t, a segm ent of length t from (x, y) to a point (s , s 2)w ill be a r a y if the segm ent i s perp en d icu lar to P at (s , s ). that

9 S ,

T h is m ean s

9s (x, y, s) =0

for such a segm ent. It follow s that f i s obtained from equation for s and substitu tin g:

by so lv in g th is la s t

£ £ = _ * + 9s 2

1 - 2y 2

s + s

T h ere a re va lu es fo r (x, y) in a neighbourhood of which we cannot so lve the equation 3 g /9 s = 0 sm ooth ly fo r s . T h ese re p re se n t points at which Wt is tangent to a light ra y determ in ing Wt . T h ese points a re called c a u s t ic s . In a neighbourhood of the c a u st ic s , it is im p o ssib le to find a sm ooth solution of the H am ilton -Jacob i equation ^ d f , d O = 1 with the given in itia l data. In R iem annian geom etry , one can in terp re t the conjugate locus of a point p a s the cau stic se t of a point so u rce of light at p.

Section 3 d e a ls with the question of determ in ing the "g e n e r ic " lo ca l g eo m etric stru c tu re of c au stic s e t s . B e fo re doing so , we broaden the context to which the an a ly s is of sec tio n 3 w ill apply.

2. HAM ILT ON - J AC OBI EQUATIONS

The sam e techniques u sed by geom etric op tics to give approxim ate so lu tions to M axw ell's equations can be u sed to give approxim ate so lu tion s to m ore gen era l p a r t ia l d iffe ren tia l equations [5, 6 ]. The approx im ate so lu tion s a re found by so lv in g a H am ilto n -Jaco b i equation for the c h a ra c te r is t ic s of the p a r t ia l d iffe ren tia l equation. With th is b r ie f m otivation, we p roceed to su m m arize the H am ilton -Jacob i theory.

We a re in terested in equations which can be w ritten lo c a lly in the form

H(x, ux) = 0

with x = (xb . . . , xn) R n , u :R n ->R, and ux = (9u/9xi, . . . , 9u/9xn). Given H :R 2n -» R, a function u :R n -*■ R such that H(x, u x) = 0 i s called a solution

28 4 GUCKENHEIMER

of the H am ilton -Jacob i equation with the H am iltonian H. L e t u s now w rite such an equation in a c o -o rd in a te -fre e m anner on an n-dim en sion al m anifold M. We in te rp re t H a s a function on the cotangent bundle T*<M of M. A solution of H = 0 i s a function u : M ->• R such that H | graph (du) = 0. H ere du : M -► T *M is the ex te r io r d eriv ativ e and it s graph is the im age of du a s a sec tio n of the cotangent bundle.

T h ere i s a s t i l l m ore geom etric rep re sen ta tio n of the so lu tions of H = 0. To p re se n t th is rep resen ta tio n , we need to in troduce the sym plectic stru c tu re of T *M . C on sid er the com m utative d iag ram :

T (T *M )

The m aps 7Г2, and 7Г3 a re bundle p ro je c tio n s; лц i s dw2. Define the canonical one form и on T *M by

u(X) = (/гз(Х))(7г4(Х))

The canonical two form Г2 on T *M is defined to be du. Í2 i s a c lo sed two form of m ax im al rank. If we choose co -o rd in ates {q i, . . . , qn} on M and then take lin ea r co -o rd in a tes {p i, . . . , pn} in each fib re of T *M re la tiv e to the b a s is {dqi, . . . , dqn}, then

nU = Z/ pidqi

i=lw here dq¿ i s now in terp reted a s a one form on T#M in stead of M. In these lo c a l co -o rd in a te s.

Cl = Zy dpi A dq¡

The re a so n for in troducing Cl at th is point i s the follow ing lem m a:

L e m m a : L et в be a one form on M, and let i:g rap h (в) -» T *M be the in clusion . Then в i s c lo sed if and only if i*f2 s 0.

P r o o f : We v e rify the lem m a in the lo ca l co -o rd in ates introduced above. In th ese co -o rd in a te s w rite в = Ea¡ dq¡ for som e a¡ : M -» R . (Note that dq¡ i s a one form on M in th is e x p re ss io n .) The graph of в is the se t of point with co -o rd in a te s (qi, . . . , qn, ai(q), . . . , a n(q)). The tangent sp ace to graph (в) i s spanned by

Г Э ул 9aj Э 1.1 9q¡ j 9q¿ 9pj i

X 2 tangent to graph (0).

Эа^ 9a¿9Qí 3qk

To evaluate we m ust com pute £2(Х ь X 2) fo r Xi,Now,

Q ( ± . + y ^ + У )V 9qt 4-, 9q t 9р; ' 9qk 9qk 9p¡ J

IAEA-SMR-11/35 285

Эа^ Э а j 9qi 9qi(

T h is i s sa t is f ie d if and only if i*f2 = 0, a s is evident from the above form ula.

P o in c a re L e m m a : A c lo sed one form в on M can be w ritten lo ca llya s du fo r som e function u : M -> R .

Consequently, we obtain the following ch arac te r iza tio n o f so lu tion s of H = 0: an n -d im en sion al subm anifold i : X -* T *M lo c a lly re p re se n ts a solution of H = 0 if

1) H I X = 02) i * Q = 03) X is t r a n sv e r se to the f ib re s of T *M .

Condition 3) gu aran tees that X is lo ca lly the graph of a one form on M;2) i s the condition for that one form to be lo c a lly exact.

We p ro p o se now to g en era lize the defin ition of a solution of H = 0 by s im p ly forgettin g condition 3).

D efin ition : A L agran gean subm anifold of T *M is an n-d im en sion al subm anifold i :X -* T *M such that i*f2 = 0. A solution of H = 0 i s a L agran gean subm anifold of T *M such that H° i = 0. The sin g u la r se t of X is the se t of points at which X is not t r a n sv e r se to the f ib re s o f T *M .

The sin gu la r se t c o n sis ts of those points of X at which X cannot be w ritten lo c a lly a s graph(du), u:M-*-R. In analogy with the definition of a light cau stic , we define the cau stic se t o f X to be the p ro jectio n of the s in g u la r se t of X into M.

In the next sectio n , we sh a ll con sid er gen eric c a u st ic s of so lu tion s.To sp eak of g en eric ity , it i s n e c e s sa ry to give the se t o f so lu tions the s tru c tu re of a topo log ical sp ac e . Within the fram ew ork we have developed, th is can be done in a stan dard way. The sm ooth em beddings of an n -d im en sion al m anifold N in T *M form an infinite d im en sion al m anifold, and those em beddings which a re L agran gean form a c lo sed subm anifold of th is in fin ite d im en sion al m anifold. If one w ish es to w ork with u n p aram etrized su bm an ifo lds, we m ay divide by the free action of the d iffeo m orph ism group of N. The se t of so lu tions of H = 0 which a re d iffeo m orph ic to N in h erits a topology from th is sp a c e . In th is topology, two su bm an ifo lds Xb X2 a re c lo se i f th ere i s a d iffeom orph ism i: Xi -» X2 which is C“ -c lo se to the in clu sion i: Xi ->■ T *M in C°°(Xi, T *M ).

B efo re p roceed in g to the d isc u ss io n of c a u st ic s in the next section , le t u s re m ark that th ere is a sligh tly d ifferen t p rob lem which a r i s e s in H orm ander [5 ] and can be dealt with in an en tire ly analogous m anner.In th is p rob lem , one obtains a H am iltonian H which is a p o sitiv e ly hom ogeneous function of the fib re co -o rd in a te s in T *M . T h is m ean s that if f GTq*M and с > 0, then H(?) = H (c f) . One then looks fo r p o sitiv e ly hom ogeneous L ag ran g ean m anifolds which, a re so lu tions of H = 0. A L ag ran g e an m anifold i s p o sitiv e ly hom ogeneous if it i s a union of r a y s in the f ib re s of T *M . Such a m anifold cannot be tr a n sv e r se to the f ib re s of T *M ; so it s c au stic se t i s the en tire im age of it s p ro jectio n into M. The

The form в i s c lo sed i f and only if

286 GUCKENHEIMER

se t of p o sitiv e ly hom ogeneous L agran gean m anifolds aga in has the stru c tu re o f an in fin ite-d im en sion al m anifold; hence, gen eric ity m ak es se n se in this context. The r e su lt s stated in the next sec tio n apply to both c a s e s .

3. CA TA STRO PH ES AND CAUSTICS

Our aim is to d e sc r ib e the lo c a l s tru c tu re of gen eric c a u stic s of so lu tion s of a H am ilton -Jacob i equation. The m ain re su lt is the following, w here "g e n e r ic " m ean s belonging to som e open-dense se t of the ap p rop ria te topo logical sp ace .

T h e o re m : F o r a gen eric H am iltonian H, the cau stic se t of a gen eric so lu tion of H = 0 h as the lo c a l stru c tu re of the product of a d isk with an e lem en tary c -ca ta stro p h e of Thom .

We now recount enough of T h om 's catastrop h e theory to define an e lem en tary catastro p h e . L e t f : Rn-> R be a function having a c r it ic a l point at 0. If 0 i s a non-degenerate c r it ic a l point, then any perturbation of f w ill have a unique, non-degenerate c r it ic a l point near 0. H ow ever, if 0 i s a degen erate c r it ic a l point of f, then a perturbation of f m ay have se v e ra l c r it ic a l points near 0 o r none at a ll. F o r exam ple, i f f: R -» R i s defined by f(x) = x5, then a perturbation of f m ay have 0, 1, 2, 3, o r 4 c r it ic a l points n ear 0. T h om 's catastro p h e theory allow s us to an a ly se the behaviour of (degen erate) c r it ic a l points under perturbation .

C o n sid er the sp ace P of p ro p er C“ functions f: Rk -» R . The groups Dk and Di of C” d iffeo m o rph ism s of Rk and R act on P by com position on the righ t and le ft, re sp e c tiv e ly . That two points of P lie in the sam e orb it of the action o f X D j m ean s that the two functions d iffer only by co-ord in ate ch an ges. The o rb its of D^ X Di give P a s tra t if ie d stru c tu re . The open o rb its a re called stab le functions. It i s a theorem of M o rse theory that the stab le functions a re those with non-degenerate c r it ic a l points and d istin ct c r it ic a l v a lu e s .

We define the codim ension of fG P to be the codim ension of its orb it under the action of D ^ X D j. L et f € P have codim ension m . Then we choose a m ap Ф : Rm -► P such th at$ (0 ) = f and such that Ф is t r a n sv e r se to the orb it of f at f. The m ap Ф is ca lled a u n iv e rsa l unfolding of f. The in te rsec tio n of the im age of Ф with the v a r io u s o rb its of Dk X Di defines a s tra t ifica tio n of R m. In p a rticu la r , con sider the se t ССФ(ЙШ) of non -stab le functions. С i s a s tra t if ie d se t with a natural s tra t ifica tio n (which m ay be d ifferen t from that given by it s in tersectio n with the o rb its of D^ X D i. ). Choose Rnc R m such that Rn is t r a n sv e r se to the stra tu m of 0. Then CDRn is called the e lem en tary catastrop h e of f. Its germ at 0 is determ ined up to iso m o rp h ism of s tra t if ie d s e t s . U su ally , one stu d ies the lo c a l problem which a r i s e s from choosing f to have a sin g le d egen erate c r it ic a l point and d istin ct c r it ic a l v a lu e s . M ather has proved that, for m S 6, the stratu m of 0G C c o n sis ts o f 0 i t s e lf and, consequently, m = n. It follow s from M ath er 's gen eric ity theorem that, up to iso m o rph ism of stra t if ie d s e t s , th ere a re only a finite num ber of e lem en tary c a ta stro p h es of a given d im ension . Thom has given nam es to the seven e lem en tary ca ta stro p h es of d im en sion s at m ost four.

A s we have indicated above, M o rse theory g iv es two conditions which a re n e c e s sa r y and su ffic ien t for the s tab ility of a function f: (1 ) f has non-degenerate c r it ic a l points only, and ( 2) no two c r it ic a l points lie on the

IAEA-SMR-11/35 287

sa m e lev e l su r fa c e of f. The catastrop h e se t o f a non -stab le function f sp li t s into two (non-disjoint) p ie c e s correspon din g to functions which have degen erate c r it ic a l points and to functions which have two c r it ic a l points with the sam e va lu e . The f i r s t p iece we c a ll the c -ca ta stro p h e se t of f; the second Thom c a lls the M axw ell se t of f. It is the c -ca ta stro p h e se t which a p p e ars in our theorem .

T h ese a re the b are a b stra c t outlines of T h om 's theory. P ra c t ic a lly , one can be much m ore sp e c ific in giving "n o rm a l" fo rm s for functions and th eir unfoldings having a given catastro p h e . B rie fly , this is done as fo llow s. One can take fo r f a polynom ial defined on Rk (k depends on the catastroph e) such that f(0) = df(0) = d2f(0) = 0. L e t I be the id ea l generated by the f i r s t p a r t ia l d e r iv a tiv e s of f in P . In o rd er for f to have finite codim ension , it i s n e c e s sa ry that the dim ension of P / I be fin ite. Choose a se t of polynom ials v0, . . . , vm which p ro je c t onto a b a s is of P / I with v0 a constant function. Then the unfolding of f i s given by

mФ (аь . . . , am) = f + £ a iVi

i=l

S ince d2f = 0, we m ay take Vj a s the i-th co-ord in ate function of Rk for1 á i s k. L a te r , we sh a ll choose a d ifferen t n orm al form m ore su itab le for app lication to the H am ilton -Jacob i theory.

L e t us look m ore c lo se ly at a fam ily of functions Ф : Rm -* P . C o r r e s ponding to Ф is a m ap F : R k X Rm R X Rm defined by F (x , t) = (Ф(t)(x), t ) . Denote Tri°F by F i where n\ : R X Rk -* R i s the p ro jectio n . The se t E of c r it ic a l points of F i s the se t of points (x, t) fo r which

- °

T h is i s the union of the s e t s of c r it ic a l points for the functions F i ( • , t) :Rk -*• R. D efine the m ap a : E-* T *R m by

tf(x ,t) = (t, | f i ( x , t ) )

P ro p o sitio n :1) F o r gen eric fa m ilie s of functions Ф :R m -*■ P , a defines a L agran gean

subm anifold of T *R m .2) If к Sm , the correspon din g m ap from g e rm s of fam ilie s o f functions

to g e rm s o f L agran gean subm an ifo lds of T *R m is su r je c tiv e .We om it the p roof o f th is propositio n (which is not d ifficu lt). T h is

propositio n g iv es us the m ean s of u sin g the catatrophe theory to study the c a u st ic s of L agran gean m an ifo lds. L o ca lly , every L agran gean m anifold a r i s e s from the se t of c r it ic a l points of a fam ily o f functions. To study the cau stic se t , one h as the follow ing:

P ro p o sitio n : L e t Ф be a fam ily of functions so that Е(Ф) i s a m anifold. Then the sin g u la r points of ct(E) (a s defined in sec tio n 2) a re the im ages of points (x, t) fo r which F ( - , t) h as a d egen erate c r it ic a l point a s x.

C o ro lla ry : The cau stic se t of cr(E) is isom orph ic to the c -ca ta stro p h e se t of Ф .

288 GUCKENHEIMER

The propositio n and c o ro lla ry im ply that the study of c a u st ic s of gen eri L agran gean m anifo lds is lo ca lly equivalent to the study o f the c -ca ta stro p h e se t s of gen eric fa m ilie s of functions. The only rem ain in g tool needed for the p ro o f of the theorem is the Thom tra n sv e r sa l i ty theorem . The t r a n s v e r sa lity theorem im p lie s that, for gen eric H am iltonians, the gen eric solution of each of th ese i s a gen eric Lan gran gean m anifold.

U sing the n orm al fo rm s for unfoldings of s in g u la r it ie s , one can w rite n orm al fo rm s for g e rm s of gen eric L agran gean m anifolds of T *R n. If X c T *M is a L agran gean m anifold through p, then one m ay choose canonical co -o rd in ate s so tfiat X is t r a n sv e r se to the constant sectio n o f T *R nat p.T h is m ean s that lo ca lly X can be w ritten a s graph (dh), w here now h :R n* -» R and we identify (Rn*)*with Rn. C on sider the fam ily

H (x, Ç) = h(?) - Z) Xi?i

for xG R n and |G R n*. We then have that

graph (dh) = | | | , Ç J

which i s a lso the se t E(H) of those (x, §) for which

Thus, in the second p art of the f i r s t p roposition above, we m ay alw ays choose the fam ily of functions to be of the sp e c ia l form of H in which the unfolding p a ra m e te r s ap p ear a s lin ear com binations of the co-ord in ate functions in the dom ain.

We can obtain n orm al fo rm s of this type for gen eric L agran gean m anifo lds correspon din g to fam ilie s of functions fo r which the stra tu m of0 in the unfolding sp ace w as an iso la ted point. To do th is, take the norm al form

nî - Z / » iV i

i=l

which we obtained b efore for f :R k ->■ R with v4 = = i-th co-ord in ate ofRk for 1 s i s k. If к = n, th is i s o f the d e s ire d form . O therw ise, if к < n, re p la c e f by g : Rn -* R defined by

Пg(xb . . . , Xn) = f ( X l , . . . , Xh) - \ Z / ( X j - V j ( x b . . . , Xk))2.

j=k+l

Then g h as the sa m e catastrop h e se t a s f and the unfolding of g is given by

g - f j а Лi=l

A s a final re m ark , we note that the theory we have d e sc r ib ed allow s one to p lace T h om 's th eory of ca ta stro p h es into a dynam ical fram ew ork without re so r t in g to h is " s t a t ic m odel" [1 1 ] . T h is approach avoids som e of the tech n ical d ifficu ltie s which seem to be re la ted to the s ta t ic m odel [3]

IAEA-SMR-11/35 289

The author would like to thank the U n iv ersity o f W arwick for kind h o sp ita lity extended during p rep ara tio n of th is p ap er.


R E F E R E N C E S

[1] ARNOLD, V .I . , Characteristic class entering in quantization conditions, Functional Anal. Appl. 1(1967) 1.

[2] DARBOUX, Mémoire sur les solutions singulières des équations aux dérivées partielles du prem ier ordre, Mémoires de l 'In s titu t Sav. Etrangers (1883).

[3] GUCKENHEEMER, J. ."Bifurcation and catastrophe", Dynamical Systems (PEIXOTO, M.M., Ed.)(1973 ) 95.[4 ] GUCKENHEIMER, J . , Catastrophes and partial differential equations, Ann. Inst. Fourier 23 (1973) 31.[5 ] HORMANDER, L ., Fourier integral operators I (especially section 3 .1 ). Acta M ath. 127 (1971) 79.[6 ] HORMANDER, L ., DUISTERMAAT, Fourier integral operators II (to appear).[7] LATOUR, F . , Stabilité des champs d ’ applications différentiables; généralisation d 'u n théorème de

J. M ather. C .R. Acad. Sci. Paris 268 (1969) 1331.[8] MATHER, J . , S tability of mappings I-VI.

I: Annals of M athem atics 87 (1968) 89.II: Annals of M athem atics 89 (1969) 254.Ill: Publ. M ath. IHES 35 (1968) 127.IV: Publ. M ath. IHES 37 (1969) 223.V: Advances in M athem atics 4 (1970) 301.VI: Proceedings of Liverpool Singularities Symposium, Springer Lecture Notes in M ath. 192 207.

[9] PORTEOUS, I . , Normal singularities of submanifolds. J. Diff. Geom. J> (1971) 543.[10] THOM, R ., S tabilité Structurelle e t Morphogenèse, Addison-Wesley.[11] THOM, R ., Topological models in biology. Topology 8 (1969) 313.[12] THOM, R . , LEVINE, H . , Lecture notes on singularities, Proceedings of Liverpool Singularities

Symposium, Springer Lecture Notes in M ath. 192.[13] WALL, C .T .C . , Lectures on С-stability and classification. Proceedings of Liverpool Singularities

Symposium, Springer Lecture Notes in M ath. 192 178.[14] WEINSTEIN, A . , Singularities of fam ilies of functions. Berichte aus dem M athematischen Forschungs-

institut 4 (1971) 323.[15] WEINSTEIN, A ., Lagrangean manifolds. Adv. M ath. 6 (1971) 329.

IAEA-SMR-11/36

THE TOPOLOGICAL DEGREE ON BANACH MANIFOLDS

C . A . S . ISNARDInstituto de Matemática Pura e Aplicadaj Rio de Janeiro, Brazil

Abstract

THE TOPOLOGICAL DEGREE ON BANACH MANIFOLDS.The degree of a proper C ^Fredholm map is defined and its invariance, homotopy and m ultiplicative

properties are proved. Some transversality theorems of independent interest are proved. The Sard-Smale theorem for proper maps follows as a corollary.

INTRODUCTION

The su b jec t of th is a r t ic le i s a definition of a d egree fo r p ro p er C ^ F re d h o lm m aps between C 1-B anach m anifo lds, extending the work of Sm ale [3 3 ] , Brow der [3 ] , Elw orthy [13] and T rom b a [1 3 ] . The re su lts obtained here a re the b a se fo r the construction , in R e fs [42 , 44 ], of a d egree fo r p ro p er continuous m aps of the type f+c, from C 1-B an ach -m an ifo ld s to su b se ts of Banach sp a c e s , where f i s a C-^-Fredholm m ap and с i s a lo c a lly com pact m ap (perhaps n on-d ifferentiab le). That d egree co in cides n u m erica lly with the L e r a y - S chauder d egree , [2 2 ] , which is the p a r ticu la r c a se w here f i s the identity on som e open su b se t of a Banach sp ace , and с i s g lobally com pact.

S m a le 's , E lw orth y 's and T ro m b a 's d e g re e s req u ired the m aps and m anifolds to be of c la s s C2 . B ro w d e r 's d egree w as defined fo r C 1-F red h o lm - m ap of index 0, but it req u ired the dom ain to be an open convex su b se t of a Banach sp ace . It w as h is su ggestio n of the non-convex c a se a s a doctora l d isse r ta tio n topic that s ta rte d th is r e se a rc h .

The c la s s of m aps between oriented C 1-m an ifo ld s fo r which one can define an in teger-v alu ed d egree i s expanded here to the " C 1-® *-m ap s"(further exten sion s can be found in Ref. [4 2 ]) .

The m ain too ls in the con struction s of the d egree a re the t r a n sv e rsa lity techniques in sec tio n s D and E , which reduce the p rob lem to finite d im en sion s, i . e . to B ro u e r 's d egree . The sam e techniques a re u sed in R efs [42, 43] to reduce from fin ite d im ensions to dim ension 1 , where degree th eory i s a tr iv ia l consequence of o rd er . Through the reduction to fin ite d im en sion s we a lso re-o b ta in h ere S a rd -S m a le 's theorem fo r p ro p er F redh olm m aps from S a r d 's re su lt [3 1 ] . The settin g of m anifo lds, ra th er than Banach sp a c e s , i s a req u irem en t of the m ethods em ployed, b ecau se the p re - im a g e of a lin ea r su b sp ace by a n on -lin ear t r a n sv e r sa l C 1-m ap w ill be a C 1-m anifold , p erh ap s non -lin ear.

The d egree of com plex d ifferen tiab le F red h o lm m aps i s a lso included, giving again som e of the r e su lts in R ef. [ 13].

A s w as pointed out in R ef. [ 33 ], the d egree of C k+1 F red h o lm m aps of index к > 0, a t re g u la r v a lu e s, i s an elem ent of the non-oriented cobo rd ism additive group r)k of k -d im en sion al com pact С m an ifo ld s defined by Thom in

291

292 ISNARD

Ref. [ 39 ]. When the С k+1-m an ifo ld s a re orien ted and the m aps a re the С к+1 -Ф *(Т )-т а р в , where T i s su r je c tiv e and has a k -d im en sion al kernel, we lik ew ise define an oriented d egree , taking v a lu e s in the correspon din g orien ted cob o rd ism group f2k. We have an oriented d egree a lso in the com plex c a se , through the in terpretation of each k-d im en sion al com plex m anifold a s a 2k-d im en sion al r e a l oriented m anifold. In the text the sta tem en ts of the d egree th eo rem s a re a ll fo r the in dex -0 c a se , but we sta te and prove a l l the a u x ilia ry th eo rem s in the m ore gen era l context, so that, by a sim p le rep lacem en t, one obtains the d egree th eo rie s of m aps with positive index.

A. THE SET TIN G FO R THE D EG R EE

X and Y a re a r b it r a r y topological sp a c e s , and X and Y a re open su b se ts of X and Y, re sp e c tiv e ly , that in the topology from X and Y, re sp ec tiv e ly , a re rea l, rea l-o r ien ted , o r com plex C 1-B anach m anifolds (definitions in se c tio n s A and B), with X a lso supposed to be a H ausdorff sp ace (X -X and Y -Y m ay corresp on d , e. g. to the boundaries of the m anifo lds, o r m ay be em pty). When G is any open su b se t of X, G i s the c lo su re of G in X and 9G = G - G. — A A

We c a ll a m ap f: X -*■ Y p ro p er if fo r ev ery com pact su b se t К of Y the se t f _1(K) is com pact. The d egree w ill be defined fo r continuous p ro p er m aps f: X -* Y that have fo r r e s tr ic t io n s C 1- $ 0-m ap s X -» Y. Section A w ill be spent in defin itions, of Ck-B an ach -m an ifo ld s (k S 1 , in teger o r +°o), and C k-®m-m ap s (m in teger).

The topo log ies on B anach sp a c e s and on th eir su b se ts w ill a lw ays be the m etr ic ones given by the n o rm s. If Eo and E a re r e a l o r com plex Banach sp a c e s , Ь (Ец, E) i s the Banach sp ace of a ll continuous lin ear o p e ra to rs T : Eo -*■ E, with norm defined by | T | = su p { | Tu | | | u | = 1 }.A F red h o lm o p era to r i s any T e L ( E 0, E) such that dim N(T) < +oo and d im (E /R (t)) < +00, where N(T) and R(T), re sp ec tiv e ly , denote the n u ll-sp ace and the ran ge of T . The index of T i s dim(N(t)) - dim (E /R (T )), and «ÊmiEo, E) i s the se t of the F red h o lm o p e ra to rs E 0 -* E of index m. If Eo o r E a re fin ite-d im en sio n al and Фт (Е 0, E) f p, then dim E 0 = m + dim E and L ( E 0, E) = Фт (Е 0,Е ) .

If X i s open in Eo and f: X -* E i s a C k-m ap such that f'(x) i s in som e su b se t a of L ( E 0, E) fo r a ll x in X, then f: X -» E is ca lled a C k-q -m ap . The С ^ Ф ^ Е о , E )-m a p s a re a lso ca lled Ск-Фт - т а р 8.

A point x e X i s ca lled a re g u la r point fo r f: X -> E if and only R(f' (x)) = E. If f is а С 1-Ф о -та р , then x i s a re g u la r point fo r f if and only if f'(x) i s a on e-to-one op erator, and if and only if f'(x) i s an iso m o rp h ism onto E, b ecau se i f T e Ф0(Е 0,Е ) , thenN (T) = 0«-> R(T) = E*-> T i s an iso m o rph ism E 0 = E * - R(T) = E .

A re a l or, re sp ec tiv e ly , com plex a t la s of c la s s C k fo r som e topologicalsp ace X i s any co llection of h om eom orph ism s a : Ua = а (и а ), ca lled thech arts of the a t la s , where

1) The Ua fo rm an open co v er of X, and each o(Ua ) i s an open su b se t of som e re a l o r com plex, re sp ec tiv e ly , Banach sp ace E a ;

2) If n Ua2 f p fo r any ch arts а-у, ot<i in the a t la s , then mapa2 aÿ1: Ua2) = ar2(Uai n Utt2) £ е „ 2 i s а С к- т а р , that we c a ll a

change of ch arts in the a t la s . If, fu rth erm ore , a ll the Ea a re equal to one

/ IAEA-SMR-11/36 293

fixed re a l B anach sp ace E f 0, and the changes of ch arts a re C k-G L ¿ (E )-m ap s, w here G L j( E )Ç L ( E , E) is to be defined in sectio n B , then the a t la s i s ca lled an orientation a t la s of c la s s C k fo r X (R em ark, where E - Rn: T e G L j(R n) *-<• the determ inan t of the m atrix rep resen ta tio n fo r T i s p o sitiv e ). Any two re a l o r com plex o r orientation a t la s , re sp e c tiv e ly , of c la s s C k fo r a topo logical sp ace X is c a lled equivalent if their union, i. e. the collection of a ll c h arts in e ith er a t la s , i s a lso a re a l o r com plex o r orientation a t la s , re sp e c tiv e ly , of c la s s Ck fo r X . A r e a l o r com plex o r oriented С k-B anach m anifold, re sp e c tiv e ly , is any p a ir co n sistin g of a topological sp ace X and of an equivalen ce c la s s of r e a l o r com plex o r orientation a t la se s , re sp ec tiv e ly , of c la s s C k fo r X. Any a t la s in the equ ivalen ce c la s s i s ca lled an a t la s fo r the m anifold, and the m anifold i s c a lled a C k-m anifold m odelled a fte r the co rresp on d in g co llection of B anach sp a c e s E a . If dim E a = m < +oo fo r a ll a ,X i s c a lled a m -d im en sio n al C k-m anifold .

Open su b se ts of Banach sp a c e s E a re alw ays co n sid ered a s С k-m an ifo ld s, m odelled a fte r E (k S ®) with the identity m ap on the se t being the only ch art in an a t la s .

If X and Y a re re a l, com plex o r oriented C k-B anach m anifo lds and f: X -» Y i s any continuous m ap, we c a ll a com position of f with ch arts at a point x e X any m ap /3 f o ' 1 : a(Ua П f _1(US)) -» E e, w here a and |3 a re c h arts in a t la s e s fo r X o r Y, re sp e c tiv e ly , Ua containing x and Ua containing f(x).We sa y that f: X -*■ Y i s a C k-m ap or а С к-Фт - т а р or a Ck-u-m ap, re sp ec tiv e ly , (for som e c la s s a of l in e a r op erato rs) if a t a l l points of X there a recom po sition s of f with c h arts that a re C k-m ap s o r C k-®m-m ap s o r C k-a -m a p s ,re sp e c tiv e ly . The C k- $ m-m ap s a re a lso ca lled Ck-F re d h o lm -m ap s of index m. If X o r Y a re fin ite-d im en sio n al, a C k-m ap f: X -> Y i s а С к-Фт- т а р ifand only if dim X = m + dim Y.

A re g u la r point fo r a Ck-m ap f: X -» Y (k s 1) i s any x e X such that a(x) i s a r e g u la r point fo r som e com position (3 • f ■ а "г of f with ch arts at x.

B . TH E SET TIN G F O R THE O RIEN TED D EG R EE

A d egree taking value in Z 2 (= in te g e rs m odulo 2) w ill be defined for a ll the m aps sa tis fy in g the gen era l conditions of sectio n A . Such degree sh a ll be ca lled a mod 2 d egree . Som etim es there i s an in teger-v alu ed degree that, taken m odulo 2, g iv es the corresp on din g mod 2 d e g re e . One such c a se i s when the m anifo lds a re com plex. The other c a se i s the su b jec t of this section : when X and Y a re oriented C 1-m an ifo ld s and the m ap f: X -> Y s a t i s f ie s som e condition that p e rm its u s to a s s ig n one positive or negative orientation to each of it s re g u la r points (in a lo ca lly constant way, coherent with fin ite-d im en sio n al t r a n sv e r sa l ity ) . We c a ll th is the oriented c a s e , and the correspon din g in teger-v alu ed d egree is the oriented d e g re e . When X and Y a re oriented C 1-m an ifo ld s m odelled a fte r R mand f: X -* Y i s a C ^ m a p , the orientation of any re g u la r point x e X i s the sign of the determ inant of the Jaco b ian at a(x) of an a r b it r a r y (3 f a " 1, com position of f with ch arts a t x. In infinite d im en sion s the situation co m p lica te s b ecau se the d eterm in an ts a re not alw ays av a ilab le .

D efinition: If E and E ! a re B anach sp a c e s , L c (E, E i) i s the se t of com pact o p e ra to rs E -» E 1( which a re the T e L (E , E j) such that C ({x e E | | x | S i } ) i s contained in som e com pact su b se t of E i . We c a ll L f(E , E i) the se t of a ll

294 ISNARD

T 6 L ( E ,E i ) with fin ite rank (rank (T) = d im R (T )). Then L f(E , E i) £LLC(E, E i ) , and those two se ts a re lin e a r su b sp a c e s of L (E , E j ) . F o r any T e L (E , E x) we c a ll T + L f(E , E j) o r T + L C(E, E j ) , re sp ec tiv e ly , the su b se t {T + С I С e L f ( E , E i ) o r L C(E, E i) r e sp e c t iv e ly } of L (E , E i ) . In Ref. [ 20] it i s shown that L C(E, E j) i s c lo sed in L (E , E j) , that if T e Фт (Е, E j) , then T + L C(E, E j) с Фт (Е, E j ) , If T j e L f E . E i ) and T2 e L ( E b E 2 ), where E, E x and E 2 a re Banach sp a c e s , then T2 T i i s com pact or, re sp ec tiv e ly , has fin ite rank if e ith er T 2 o r T i i s com pact or, re sp ec tiv e ly , has finite rank.

Definition: If T e L (F , F ) w here F / 0 i s fin ite-d im en sio n al, we define det(T), the determ inant of T , to be the determ inan t of any m atrix rep resen ta tio n of T obtained by choosing a b a s is fo r T . The value of det(T) i s independent of the chosen b a s is I: E -* E w ill alw ays denote the identity o p erato r.D efinition: If T e I + L f (E, E), where E f 0 i s any Banach sp ace , le t F / 0 be any fin ite-d im en sio n al lin e a r su b sp ace of E containing the ran ge of T -I.Then T ( F ) £ F and we define det(T ), the determ inan t of T, to be the d e te rm inant of the re s tr ic t io n F -> F of T . T o show that det(T) i s iridependent of the choice of F , suppose F 0 = R (I-T) / 0. Since F q Ç F , it follow s that it su ffic e s to p rove the statem en t when E it se lf i s fin ite-d im en sio n al. Then E = F x E 0 fo r som e Eq, so it su ffice s to prove that if F and E o a r e f in ite d im ensional, and T 6 L (F x E 0, F x E 0) i s of the type T(u, v) = (Tu + Sv, v) fo r som e T e L (F , F ) and S e L ( E 0, F ), then det T = det T . T h is follow s im m ediate ly from p ro p e rtie s of determ in an ts.

P ro p o sitio n : If T j , T2 £ I + L f (E, E) then T jT 2 £ I + L f (E, E) anddet(T 1 T 2 ) = det (Tx ) • det (T 2).

P ro o f: T ake F / 0 containing R (T j- I ) + R (T 2-I), F a fin ite-d im en sio n al lin e a r su b sp ace of E, then R (T 1 T 2 -I) £ F , and co n sid er the re str ic t io n s F - * F of the o p e ra to rs .

D efinitions: G L(E) = ( T e L (E , E) | T : E = E i s iso m o rp h ism },G L f (E) = G L(E) n (I + L f(E, E)), G L C(E) = G L(E) n (I + L C(E, E)).

P roposition : If E / 0 then G L f (E) = { T e I + L f(E, E) | det T f 0 }.

P ro o f: Suppose T 6 I + Lf(E, E), then R (T -I) £ F f 0, F fin ite-d im en sio n al £ E . C all T : F -* F the re str ic t io n of T , then det(T) = det(T), so it su ffice s to prove that T 6 G L (E )*- *T £ G L (F ) , Since dim F < +00, we get fro m R ef. [ 20] that E = E 0 ® F fo r som e E 0. So E = E 0 x F , hence it su ffice s to prove:

L em m a 1; Suppose F x, F 2, E 0 a re Banach sp a c e s and T 6 L (F j x E 0, F 2 x E 0) i s of the type T( u, v) = (Tu + Sv, v) fo r a ll u s F j and v e E o , fo r som e T : F i -* F 2 and S : E 0 -* F 2 . Then T i s iso m o rp h ism F j x Ец = F 2 x Eo Ï i s iso m o rp h ism F j = F 2 . A lso T e ®p(Fj x E 9, F j x E 0) ” T e * p ( F i , F 2).

P roof: T = Ho (T x I), w hereJT x I ê (F j x E 0, F 2 x E 0) and H e L (F 2 x E 0, F 2 x E 0) a re defined by (T x l)(u , v) = (Tu, v) and H(w, v) = (w + Sv, v). C la im that H e G L(F2 x E 0). In fac t, H = I + A, where A: F 2 x E ( - * F 2 x E 0 i s defined byA(w, v) = (Sv, 0), and fro m A о A = 0 we get (I + А) о (I - A) = I = (I - А) о (I + A).

IAEA-SMR- 11/36 295

So T: F 2 x E 0 = F 2 x E 0 i s iso m o rp h ism *-* T x I: F 1 xE o = F 2 x Eo i s i s o m orph ism «-* T: F j = F 2 i s iso m o rp h ism . A lso , dim N(T) = dim N (T xI) dim N (T),

dim F 2 x Eo F 2 x Eo _ F 2R(T) - dim r (îji x I) - d lm R (^)

P roposition : When E f 0 i s a re a l Banach sp ace then G L f (E) = G Lf(E ) и G L j(E ), where:

D efinition: GLf(E) or, re sp e c tiv e ly , GL}(E) = { T e i + Lf(E, E) | det(T) > 0, det(T) < 0 }.

To define an orien ted d egree , we need to have orien tation s a ss ig n e d to the re g u la r points of the m ap. T h ose orien tation s a re given by the d er iv a tiv e s of the com po sition s with ch arts , so those d eriv a tiv e s m ust be in som e su b se t a of L (E , E) such that ст П G L(E) h as two path-com ponents, one containing the identity I, the e lem en ts of which a re ca lled o rien ta tio n -p re se rv in g , the other com ponent co n sistin g of the o p e ra to rs that a r e called o rien ta tio n -rev e rsin g .

We cannot take a = Ф0(Е, E) o r ст = L (E , E), b ecau se in those c a s e s a n G L (E ) = G L(E ), and G L(E) has only one path com ponent when E i s any in fin ite-d im en sion al H ilbert sp ace [1 9 ] , o r i s üp o r Co [ 26 ].

It is well-known [3 7 ] , [42] that G Lf(E) and G L C(E) have two path- com ponents each, which a re G Lf(E) and G Lf(E ), and re sp . GLc(E) and GL'C(E), those la s t two defined by G Lf(E) Q G L*(E) and G Lj(E ) ç g l ' J E ) . We introduce a la r g e r se t with two path-com ponents, Ф*(1) n G L(E ), an open su b se t of L (E , E), where Ф*(1) w ill be defined next.

If Eo and E a re Banach sp a c e s and T: Eo -* E i s a F redh olm o p erato r we define Ф ^Т ), the F redh olm s t a r of T to be the se t of a ll S such that tS + ( l- t )T i s a F red h o lm o p era to r E o -» E when 0 S t S 1. Ф*(Т) i s then as ta r - l ik e se t of F red h o lm o p e ra to rs . If S e Ф*(Т), then T + L c(Eo, E ) ç ï , ( T ) . Ф„,(Т) is an open su b se t of L (E , Ex), b ecau se so a re a l l the s e t s Фт (Ео, E), by section E, lem m a 1.

F o r any re a l Banach sp ace E / 0, each se t in the in clusion G Lf(E) ç GLC(E) С Ф*(1) n GL(E) h as ex ac tly two path-com ponents (Appendix). (Actually, in Ref. [4 2 ] , it i s shown that those in clu sion s a re hom otopy eq u iva len ces, so that, when E i s any in fin ite-d im en sion al re a l Banach sp ace , a ll those s e t s have the hom otopy type of GL(°o), with hom otopy groups given by the Bott p erio d ic ity theorem . )

The se t s of orientation p re se rv in g or, re sp ec tiv e ly , orientation re v e r s in g iso m o rp h ism s, that a re , by definition, the path-com ponents of Ф*(1) n G L(E ), a re open in L (E , E ).

When E f 0 i s a fin ite-d im en sio n al re a l Banach sp ace ev ery iso m o rp h ism gets an orientation c h arac te r ize d by the sign of the determ inant, b ecau seI + L f (E, E) = L (E , E) = Ф0(Е, E) = Ф„(1) and G L f(E) = G L (E ).

к кD efinition: А С -Ф ^ -т а р i s а С -Ф „(1 )-тар . The oriented d egree w ill be defined fo r p ro p er С1- Ф * - т а р з between oriented C 1-m an ifo lds m odelled a fte r the sam e re a l B anach sp ace E (oriented m anifolds a re defined in sectio n A). F u rth e r gen era liza tio n s can be found in R ef. [4 2 ] .

296 ISNARD

D efinition: L e t f: X -► Y be а С ^-Ф ^-тар and X and Y be oriented C 1-m an ifo lds m odelled a fte r the re a l Banach sp ace E f 0. We s a y that x e X is a re g u la r point fo r f with p o sitive or, re sp ec tiv e ly , negative orientation when there a re ch arts a and /3 in a t la s e s fo r X or, re sp ec tiv e ly , Y, with x in Ua and f(x) in Ue, such that )3fa 1 h as fo r deriv ative in a(x) an orientation p re se rv in g or, re sp e c tiv e ly , re v e r s in g iso m o rp h ism E = E . T hat each re g u la r point has a uniquely defined orientation , independent of the chosen ch arts , follow s from :

M ultiplication law of orien tation s: L e t E / 0 be any re a l Banach space ,T e Ф*(1) n G L(E ), and U 6 G LC(E). Then TU and UT a re o rien ta tio n -p reserv in g if T and U have the sam e orientation , and TU and UT a re o r ien ta tio n -rev e rsin g if T and U have d ifferen t orien tation s.

Note 1: T h is law does not hold fo r u n re stric ted T and U in Ф^Ш. In fact, it i s e a sy to show that when G L(E) is path-connected, a s it i s the c a se with ip , with Co, and with a ll in fin ite-d im en sion al H ilbert sp a c e s , then any iso m o rp h ism E = E i s equal to the com position of fin ite ly m any orientation - p re se rv in g iso m o rp h ism s in Ф*(1).

P ro o f: TU and UT e Ф*(1), b ecau se if C e L C(E, E) then TC and CT e L C(E, E) (from R ef. [2 0 ]) . T h ere a r e continuous paths t -» Tt in G L(E) n Ф^(1) and t -* Ut in G L C(E), connecting T , (or U, re sp ec tiv e ly ) to e lem ents of G Lf(E ).Then t -* T t U t and t -» Ut T t a r e continuous paths in G L(E) n Ф„.(1). Since orien tation s a re constant along those paths, it su ffice s to prove the m u ltip lication law when T and U a re both in G L f(E ). In th is c a se , it i s a consequence of det(TU) = det(T) . det(U) = det(UT).

The o r ien ta tio n -p re se rv in g C k-m ap s a re the Ск-Ф ,,.-тарз (k s 1 ) fo r which ev ery re g u la r point h as p o sitive orientation . In c a se X and Y a re open su b se ts of R n, th is m eans that det(f'(x)) ê 0 at a ll x e X.

An o rien ta tio n -p re se rv in g С k-d iffeom orph ism is any orientation - p re se rv in g C k-m ap that i s a lso C k-d iffeom orph ism . The in v erse m ap is a lso an o rien ta tio n -p re se rv in g C k-d iffeom orph ism , b ecau se if T i s in Ф*(1) n G L(E ), so i s T " 1, and then T and T "1 have the sam e orientation s:In fact, i f 0 И И then T ' 1 (tT + (l-t)I) = t l + ( l- t )T " 1 e $ o (E ,E ) , hence T 1e Ф*(1), and the m ap T *-* T "1 i s a hom eom orphism G L(E) п Ф„.(1) == G L(E ) n Ф*(1), so it m akes path com ponents correspon d to path com ponents.

Re m ark : It i s e a sy to show that Ф*(1) contains I+H w henever H e L (E , E) has e s se n tia l sp e c tra l rad iu s Pe(H) < 1 (pe(H)) i s the sp e c tra l rad iu s of H + L C(E, E) in the Banach a lg e b ra L (E , E ) / L C(E, E), i. e. pe(H) = lim 1 Hn|c , where IT |c = |Т + L c(E, E) I = inf { IT + С I I C e L C(E, E ) } . It i s a lw ays true that Pe(H) S | h | c - |Н |) .

C. DEFIN ITIO N S OF TH E D EG R EE AND A PPLICA TIO N S

In a ll the defin itions of the d egree in this section , X and Y a re open su b se ts of X and Y, re sp e c tiv e ly , which a re a r b it r a r y topo logical sp a c e s .The topo logies on X and Y a re alw ays those which they rece iv e from X and Y, re sp e c tiv e ly , and the topology on X is alw ays supposed to be of the H ausdorff type. We have a continuous p ro p er m ap f: X -♦ Y and we suppose that one of the follow ing three conditions i s sa tis f ie d :

IAEA-SMR-11/36 297

(i) The gen era l c a s e : X and Y a re a r b it r a r y C1-B an ach -m an ifo ld s and the re s tr ic t io n s X -*• Y of f i s а С 1-Ф о -та р . The d egree i s an in teger m odulo 2.

(ii) The com plex c a s e : X and Y a re com plex C1-B an ach -m an ifo ld s and the re s tr ic t io n X -> Y of f i s a С ^-Ф о -тар . The d egree i s a non-negative in teger.

(iii) The oriented c a s e : X and Y a r e oriented C1-m an ifo ld s m odelled a fte r the sam e r e a l Banach sp ace E f 0, and f: X -* Y i s а С 1- Ф *- т а р .The d egree i s an in teger.

R em ark : When X and Y a re m odelled a fte r the sam e fin ite-d im en sio n al sp ac e , any C 1-m ap X -* Y i s а С1-Ф о -та р and i s а С1- Ф *- т а р .

D efinition: A point of X i s a c r it ic a l point fo r f i f it i s not a re g u la r point.A c r it ic a l value fo r f i s the im age by f of a c r it ic a l point. The re g u la rv a lu e s fo r f a re the e lem en ts of Y - f(3X) that a r e not c r it ic a l v a lu e s. So y e Y i s a re g u la r value fo r f i f and only if ev ery point of f "1 (y) i s in X and i s a re g u la r point fo r f. In p articu la r , y i s a re g u la r value fo r f whenever f "1 (y) i s em pty.

The d egree of f on X a t y, w ritten deg(f, X, y), w ill be defined at a ll y e Y - f(9X).

Definition: Suppose y e Y - f(9X) i s a re g u la r value fo r f such that f _1 (y) h as m elem en ts (m < +°o). Then we define:

(i) In the gen era l c a se , deg(f, X, y) = m (m od 2)(ii) In the com plex c a se , deg(f, X, y) = m

(iii) In the oriented c a se , deg(f, X, y) = p-n,

w here pan d n , re sp e c tiv e ly , i s the num ber of points of f " 1 (y) that a re reg u lar fo r f with p o sitive (negative, re sp ec tiv e ly ) orientation .

T h eorem 1: If y e Y — f(3X) i s a re g u la r point fo r f then f _1(у) i s a finite se t, with a num ber of e lem en ts equal, m odulo 2, to deg(f, X, y), and у h as a neighbourhood co n sistin g of re g u la r v a lu es y' e Y — f(3X) fo r f such that deg(f, X, y') = deg(f, X, y).

P ro of: We need in itially :

In v erse m apping theorem : Suppose x i s a re g u la r point fo r а С к-Ф0- т а р or, re sp e c tiv e ly , а С ^-Ф ^-тар f: X -* Y (kS 1), where both X and Y a re re a l o r com plex Ck-B an ach -m an ifo ld s or oriented Ck-B anach m anifo lds, re sp ec tiv e ly . Then x i s an som e open su b se t W in X sucn that the re s tr ic t io n of f to W is a С -d iffeom orp h ism onto som e open su b se t f(W) of Y and that ev ery point of W is re g u la r fo r f, with the sam e orientation a s x, in the oriented c a se .

P ro o f: T ak in g com position s with ch arts, we m ay suppose that X and Y a re open su b se ts of Banach sp a c e s E and E 1# f'(x) i s an iso m o rp h ism E = E j ,so x i s in som e open se t X 0 co n sistin g of re g u la r points fo r f, which a ll have the sam e orientation a s x, in the orien ted c a se , b ecau se the se t of a l l i s o m o rp h ism s E = E (o rien tatio n -p re serv in g and - re v e rs in g , re sp ec tiv e ly ) is open in L (E , E), fro m R ef. [ 20] (or fro m Section B, re sp ec tiv e ly ). We apply now, to f r e s tr ic te d to X Q, the well-known in v erse m apping theorem [2 0 ] .

298 ISNARD

P ro o f of th eorem 1: Suppose f -1 (y) f 0. E ach point of f " 1 (y) is in som e open se t W in X a s in the statem en t of the in v e rse m apping theorem , f ' 1 (y) i s com pact and, th erefo re , covered by fin ite ly m any of those s e t s W, say W1( . . . , Wq , each containing som e point of f ' 1 (y). Since f i s one-to-one on each o f those s e t s W^, it contains exactly one point of f ‘ 1 (y). Then f 'Ну) h as ex ac tly q e lem en ts. We m ay suppose that the W\ a re p a irw ise d isjo in t, taking them sm a lle r , if n e c e ssa ry . C all now V = ( (V f(W \)) - f(X - A w *)- Then y £ V , and if y 'e V, f 1(y') has q points, one on each se t Wx . F ro m this we obtain: y' i s a r e g u la r value fo r f, y 'E Y - f(9X), and deg(f, X, y 1) == deg(f, X, y). We m ust now show that V i s an open su b se t of Y. We take c a re of the c a se f _1 (y) = fi showing that Y — f(X) i s a lso open in Y. The p roof i s then com pleted by

L em m a 1: L e t f be any continuous p ro p er m ap from som e topo logical sp ace to and le t Y be an open su b se t of Ÿ and a Ck-B an ach -m an ifo ld (кй 0) in the induced topology fro m Ÿ. Then, fo r any c lo sed su b se t A of the dom ain of f, we have Y — f(A) open in Y and Y n f(A) c lo sed in Y.

P ro o f: Since Y is lo c a lly m etrizab le , it su ffic e s to show that if any sequence f(an) con v erges in Y to y, with ane A , then y € f (A ) . The se t con sistin g of y and of the f(an) is com pact, so i t s p re - im a g e by f i s a com pact se t, which contains the a n. The an have then a gen era lized subsequence converging to som e x, which m ust be in A, b ecau se A i s c lo sed . By continuity, f(x) = y, hence y e f(A).

Definition: If y e Y - f(9X) i s a c r it ic a l value fo r f, define deg(f, X, y) == lim deg(f, X, yn), w here (yn) is an a r b it r a r y sequence of re g u la r v a lu es fo r f, converging to y in Y - f(9X). T h is lim it i s w ell defined, in the sen se that deg(f, X, y n) i s constant fo r a l l n S n 0, and th is constant value does not depend on the chosen sequence, b ecau se of:

T h eorem 2: Suppose y 6 Y - f(9X). Then y i s in som e open se t V inY - f(9X) such that deg(f, X, y) i s constant fo r a ll y in V re g u la r value fo r f.

S a rd -S m ale T heorem : E v e ry point y e Y - f(9X) i s the lim it of som e sequence of re g u la r v a lu e s .

T h ose th eo rem s a re proved in sectio n D. A s consequence we obtain:

In varian ce property : deg(f, X, y) i s constant fo r y in any fixed connected com ponent of Y - f(9X).

P ro o f: We have deg(f, X, y) constant fo r any y in the se t V given by theorem 2. So the d egree i s a lo c a lly constant function of y, hence it i s constant on connected su b se ts of Y - f(9X).

S u r jectiv ity property : If deg(f, X, y) f 0 fo r som e y e Y - f (ЭХ), th e n y e f (X ) .

P roof: If y ¿ 'f(X ) then y i s a r e g u la r value .

D efinition: L e t now G be any open su b se t of X. The re str ic t io n G -* Ÿ of f isp ro p er, and has fo r re s tr ic t io n а С1-Ф 0- т а р or, re sp ec tiv e ly , а С 1- Ф *- т а р

IAEA-SMR-11/36 299

G -* Y, w here G is the re a l or com plex, or, re sp ec tiv e ly , oriented C 1-B an ach - m anifold obtained by re str ic t io n of ch arts . We w rite deg(f, G, y) fo r deg(f/G , G, y). We have then a lo c a l d egree defined.

A dditiv ity p rop erty : If y e Y - f(X - U Gx), w here the Gx form a fin ite or infinite co llection of open su b se ts of X, then deg(f, X, y) = Ç deg(f, G^, y), and th is sum i s fin ite, i. e. deg(f, Gx, y) f 0 fo r a t m ost fin ite ly m any in d ices X.

P ro o f: f _:L(y) i s com pact and contained in U G x, so it i s contained in fin ite ly m any of the Gx. F o r the rem ain in g Gx, we have f ’ 1 (y) n Gx = 0, hence deg(f, G x, y) = 0. So we m ay su ppose the collection of the Gx i s fin ite . At a re g u la r value, the theorem follow s from the definition of d egree . The c r it ic a l v a lu es y a re the lim it of seq u en ces yn of re g u la r v a lu e s fo r f, which we m ay suppose to be in Y - f(X - U Gx), b ecau se this i s an open se t, by lem m a 1 . Then deg(f, X, yn) = Yj deg(f, Gx, y n). and we take the lim it whenП -* +00. x

E x c isio n p ro p erty : If A i s a c lo se d su b se t of X such that y ^ f(A ) , then deg(f, X, y) = deg(f, X -A , y), w henever y e Y - f (8X).

P ro o f: T h is i s the add itiv ity p ro p erty with only one Gx, equal to X-A .T h ese p ro p e rtie s p erm it us to gen era lize a theorem of C acciopo li [ 5] to the following:

T h eo rem : L e t f: X -» Y be a p ro p e r С1-Ф()- т а р , and su ppose that Y is connected and contains a re g u la r point y fo r f such that f _ 1 (y) has an odd num ber of e lem en ts. Then Y = f(X ). The theorem has a stro n g e r fo rm if X and Y a re com plex C 1-B an ach -m an ifo ld s, o r if f i s o rien ta tio n -p reserv in g , with X and Y oriented . In those c a s e s , i f Y i s connected and contains som e re g u la r value y such that f _1(y) f 0, then Y = f(X).

P ro o f: Since ЭХ = 0 and Y is connected, the in varian ce p ro p erty g iv es deg(f, X, y) i s constant fo r y in Y. F o r the re g u la r value y in the statem ent of the theorem , we have deg(f, X, y) f 0. Then, by the su r je c tiv ity property , ev ery point of Y i s in f(X).

C o ro llary : If Y i s connected, X and Y a re com plex or, re sp ec tiv e ly , orien ted C1-B an ach -m an ifo ld s and f: X -» Y i s a p ro p er С 1-Ф0- т а р or, re sp ec tiv e ly , a p ro p er o rien ta tio n -p re se rv in g C 1-m ap, then e ith er f(X) = Y o r f(X) h as no in te r io r points.

P ro o f: B y S ard -S m ale any in te r io r point w ill be the lim it of som e sequence of r e g u la r v a lu e s fo r f. Som e of those re g u la r v a lu es w ill then be in f(X), and we m ay apply the la s t theorem to them .

D. TH E MAIN TH EO REM IN IN FIN ITE DIMENSION

We sh a ll now reduce the p roof of theorem 2 and of S a rd -S m a le 's T heorem to th e ir fin ite-d im en sio n al fo rm s, which a re the in varian ce of B ro u e r 's d egree [ 32, 42, 43] fo r C1-m ap s and S a r d 's theorem [ 31, 36 ]. We ob serve in itia lly that we m ay suppose that Y is an open su b se t of som e Banach sp ace

30 0 ISNARD

E, and that ЭХ = 0. In fact, we rep lace f by i?f: f -1 (U) -> jS(U), where U is an open su b se t of the dom ain of som e lo c a l ch art (3 fo r Y, and J S U £ Y - f(3X).

O b serve that if V £ Y £ E i s an open ball, then if y 0 and y j a re in V we have [ y0 , y J £ V с Y, where [yo, y i ] i s defined to be the c lo sed in terval { ty j + ( 1 - 1)y0 I 0 S t a 1} . So we need only to prove:

T h eorem 1: Suppose X is a re a l o r com plex or, re sp ec tiv e ly , an oriented C ^-B anach-m anifo ld (kê 1), Y is an open su b se t of som e re a l o r com plex Banach sp ace E, and f: X -» Y is a p ro p er Ск-Ф р- т а р , where p is any in teger. Then:

Sard-Sm ale-Q uin n : If к > p then ev ery у 6 Y i s the lim it of som e sequence of re g u la r v a lu es.

Main theorem fo r d egree theory: If f i s а С ^ Ф о - т а р , or а С ^-Ф ^-тар in the oriented c a se , and y 0 , уг a re re g u la r v a lu e s fo r f such that [ y o , yi 1 £ Y, then deg(f, X, y 0) = deg(f, X, y x).

P r o o fs : C a ll К = f " 1(y), o r, re sp e c tiv e ly , К = f"1 ([ y0 > Y iD . then К i s com pact. By theorem 2, which follow s, there is som e fin ite-d im en sio n al su b sp ace F of E such that R (f'(x)) + F = E fo r a l l x e K, where

D efinition: F o r ev ery x E X c a ll R(f'(x)) the ran ge of (fû'"1)'(a(x)), where a i s an a rb itr a ry ch art in som e a t la s fo r X such that x e Ua . Of co u rse , R(f'(x)) i s a lin e a r su b sp ace of E that does not depend on a.

T h eorem 2: L e t f: X -» E be а С к-Фр- т а р (p any in teger, к S 1), w here E is a r e a l or com plex Banach sp ace and X i s any rea l, com plex or oriented Ck-B an ach -m an ifo ld . Then, fo r ev ery F lin ear su b sp ace of E , the se t XF = { x e X I R (f'(x)) + F = E } i s open in X . And fo r ev ery К com pact su b se t of X, there i s som e fin ite-d im en sio n al F £ E such that R ( f '(x ) )+ F = E fo r a ll x in K.

P ro o f: T o show X F i s open we com pose with c h arts and then we m ay suppose X i s an open su b se t of som e Banach sp ace E 0. F ro m the continuity of f 1: X -» L (E 0, E) it su ff ic e s to show that the s e t of a l l S e Фр(Ео, E) such that R(S) + F = E i s open in L (E q, E), which is proved in lem m a 1 of section E.

Now, su ppose К i s com pact in X . Then each point of X i s in X F fo r som e fin ite-d im en sio n al su b sp ace F of E b ecau se fo r each Fred h o lm op erato r T fro m a Banach sp ace to E there i s som e fin ite-d im en sio n al F £ E such that R(T) + F = E (lem m a 1 of section E ). The com pact se t К i s then covered by fin ite ly m any open s e t s X F ¡, with dim F ¡ < + » . When we c a ll F the sum of the F¡ we have dim F < +oo and R(f'(x)) + F = E fo r a ll x in K.

Continuation of proof of theorem 1: We m ay suppose that the f in ite - d im ensional su b sp ace F of E given by theorem 2 contains y or, re sp ec tiv e ly , y 0 and у i, adding to F the lin e a r span of those points, if n e c e ssa ry . We m ay ac tu a lly suppose that R (f’ (x)) +■ F = E fo r a ll x e X, b ecau se by theorem 2 the se t X F of the x £ X such that R (f'(x)) + F = E i s open in X, th erefo re , we m ay rep la ce Y by Y 0 = Y - f (X - X F) and X by Xo = f " 1(Y0) £ XF (Y0 i s an open su b se t of Y by section C, lem m a 1). We have then f t r a n sv e r sa l to F , accord in g to the follow ing definition:

IAEA- SMR- 11/36 301

Definition: L e t F Q E be an in clusion of r e a l or, re sp ec tiv e ly , com plex Banach sp a c e s , and X be a r e a l or, re sp ec tiv e ly , com plex C k-B an ach - m anifold. We sa y that a C k-<6q-m ap f: X -* E i s t r a n sv e r sa l to F when R (f '(x)) + F = E fo r a l l x e f ^ F ) .

When th is condition is sa t is f ie d we have:

T ra n sv e r sa lity theorem (section E): T h ere i s som e a t la s m aking f _1 (F) into a r e a l or, re sp e c tiv e ly , com plex Ck-B an ach -m an ifo ld of d im ension q + dim F , in such a way that the re s tr ic t io n f: f _:L(F) -> F i s a C k- $ q-m ap that has a s re g u la r points exactly the points of f _:L(F) that a r e re g u la r fo r f: X -> E . If, fu rth erm o re , f i s а Ск- Ф *- т а р and X i s oriented , then there i s one a t la s m aking f _1(F) into an oriented C k-m anifold m odelled a fte r F , in such a way that the re s tr ic t io n f: f _1 (F) -> F of f is а Ск- Ф *- т а р , that has the sam e orientation at each re g u la r point a s f: X -> E.

Continuation of p roof of theorem 1: It follow s that the re g u la r v a lu e s fo r the r e s tr ic t io n f: f _1 (F) -► Y n F of f a re exactly the re g u la r v a lu e s fo r f: X - Y that a r e in Y n F . Hence S a rd -S m a le 's theorem fo r f fo llow s from the corresp on d in g re su lt fo r f. The sam e can be sa id about the m ain theorem , b ecau se [ yo, y il £ Y n F and f and ?" have the sam e d egree at each re g u la r value in Y n F (the sam e p re - im a g e s , with the sam e orien tation s in the oriented case ).

R em ark : In finite d im en sion s the com plex c a se of the Main T h eorem follow s fro m the r e a l oriented c a se : E v e ry com plex norm ed sp ace E co rre sp o n d s to a r e a l norm ed sp ace E r equal to E except fo r the m ultip lication , which is r e s tr ic te d to re a l s c a l a r s . E ach b a s is v j , . . . , vm fo r E co rresp o n d s to som e b a s is v 1( ivx, . . . . vm, ivm fo r E r. E v e ry T e L (E , E j ) co rre sp o n d s to som e T r e L (E r, E lr) with the sam e norm , and R(T) = E <-> R (T r) = Er. If T e L (E , E) then det T = | det T r |2 , a s one can see by a com putation in R ef. [ 6, p. 47] (done a fte r choice of a correspon din g b a s is such that T had a tr ian g u la r rep resen ta tio n ). Since the n o rm s in E and E r a re the sam e, C1-m ap s from open su b se ts of E to E becom e C 1 m aps from open su b se ts of Er to E r, with the d e r iv a tiv e s correspon din g through the m ap T -> Tr from LÍE, E) to L (E r, E r). So the re g u la r v a lu e s a re the sam e fo r both m aps. By

induction, we obtain the sam e statem en t fo r C k-m ap s. T : E ^ E i s an iso m o rp h ism if and only if T r : E r = E r i s an o rien ta tio n -p re se rv in g i s o m orph ism b ecau se det(Tr) = | det t | 2. Hence, if X i s a com plex Ck-m anifold m odelled a fte r E and we rep lace E by E r in a l l ch arts , X becom es an oriented C k-m anifold X r m odelled a fte r E r. And ev ery C k-m ap: X -* E b ecom es an o r ien ta tio n -p re se rv in g C k-m ap X t -» E r.

E . TR A N SV E R SA LITY

Definition of subm anifo lds: Suppose Y i s a re a l, com plex o r oriented Ck-B an ach -m an ifo ld , and su ppose a su b se t M of Y i s covered by the dom ains Ug of som e ch arts /3: Us s (3(Ug ) £ Eg in som e a tla s fo r X, such that fo r each of those 0 there i s a c lo sed lin e a r su b sp ace F s of the Banach sp ace E e such that (3(Ug n M) = /3(UB) n F 0. Then the re s tr ic t io n s ¡3: Ug n M = |3(Ug) n F e fo rm an a t la s of c la s s Ck fo r the topo logical sp ace M, in the topology from X, m odelling M a fte r the Banach sp a c e s F g . We c a ll then M a subm anifold of X,

302 ISNARD

and we sa y that those ch arts (3 m odel the in clusion M c Y a fte r the in clu sion s Fg £. Ee . If, fu rth erm ore , that a t la s fo r M happens to be an orientation a t la s , which, of co u rse , re q u ire s a ll the Fg to be equal to one fixed F , then M is c a lled an oriented subm anifold of X.

R em ark: When a ll Fg = 0 we have a О-d im en sion al subm anifold. We sh all co n sid er a ll О-d im en sion al subm anifo lds of oriented C k-m an ifo ld s to be oriented su bm an ifo lds, by definition.

T h eorem 1: If f: X -» Y i s a Ck-m ap such that f(M j) £ M2, where X and Y a re com plex, r e a l o r oriented Ck-B an ach -m an ifo ld s and Mi and M2 a re (oriented o r not) subm an ifo lds of X or, re sp ec tiv e ly , Y, then the re str ic tio n M j -» M2 of f is a lso a Ck-m ap.

P ro o f: T akin g com position s with lo ca l ch arts , it fo llow s from the p a rticu la r c a se w here X and Y a re open su b se ts of Banach sp a c e s and Mx and M2 a re in te rsec tio n s of X and, re sp ec tiv e ly , Y with c lo sed lin e a r su b sp ac e s .

D efinition: Suppose f: X -> Y is any C k-F red h o lm -m ap , with к i 1, where X and Y a re both re a l, com plex or oriented C k-B an ach -m an ifo ld s, and suppose M i s a subm anifold (oriented o r not) of Y. We sa y that f is t r a n sv e r sa l to M on a su b se t A of X when fo r ev ery x e A n Г 2(М) there is som e ch art |3 m odelling the in clusion M £ Y a fte r the in clusion Fg £ Eg of Banach sp a c e s , in som e a t la s fo r Y, such that f(x) e Ug and (|3 f)'(x) + Fg = Eg [ 1 ] .

R em ark : Any C k-F red h o lm -m ap i s then t r a n sv e r sa l to any of it s re g u la r va lu e s (with each value co n sid ered a s a 0-d im en sion al subm anifold).

The next two th eorem s w ill be proved in this section (for the d egree theory of С 1- Ф *- т а р (index 0) it i s enough to con sid er the c a se T = I, E 0 = E).

P u ll-b ack theorem : L e t X and Y be oriented C k-B an ach -m an ifo ld s m odelled a fte r r e a l Banach sp a c e s E 0 f 0 or, re sp ec tiv e ly , E , suppose f: X -> Y i s a С к-Ф „.(Т )-тар , where T: Eo~* E i s any Fred h o lm o p erato r. Then there i s one unique oriented Ck-m anifold X T m odelled a fte r E , obtained through the attachm ent of som e orientation a t la s to the topo logical sp ace X, such that the identity X = X T is an o r ien ta tio n -p re se rv in g Ck-d iffeom orph ism and f: X -» Y i s a Ck-(T + L c(Eo, E ))-m ap , i. e. the d e riv a tiv e s of the com position s of f with c h arts in a t la s e s fo r X j and Y a re o p e ra to rs of the type T + com pact lin e a r . Definition: X T i s the (T )-p u ll-b ack of Y by f. An a t la s fo r X T, which we c a ll the pu ll-b ack a t la s , c o n s is ts of a ll o rien ta tio n -p reserv in g Ck- d iffeo m o rph ism s ip between open su b se ts U,, of X and i»(U^) of Eq such that j3fcp'1 i s of the type T + m ap into som e fin ite-d im en sio n al su b sp ace of E, fo r ch arts 3 in som e a t la s fo r Y such that f(Uv) с Ug.

R em ark : When T = I, E 0 = E, X j co rre sp o n d s to the p u ll-b ack G L C- s tru c tu re s in R ef. [1 3 ] , except that now it has a uniquely defined orientation .

R em ark : Xx = X if X and Y a re fin ite-d im en sio n al b ecau se then, if T € L (E 0 , E), we have T + L C(E0, E) = L (E o , E) = Ф*(Т); hence any C k-m ap X -* Y i s а С к- Ф *(Т )- т а р and is a C k-(T + L c(E 0, E ))-m ap .

IAEA- SMR-11/36 303

(i) T h o m 's theorem : Suppose X and Y a r e re a l o r com plex C k-B anach m anifo lds (k È 1) and f: X -> Y i s a C k- $ q-m ap t r a n sv e r sa l to som e subm anifold M of Y (q any in teger). Then f _1 (M) is a subm anifold of X of dim ensionq + dim M such that the r e s t r ic t io n ? ; f " 1 (M) - M of f i s а Ск-Фч- т а р .

(ii) Suppose X and Y a re oriented Ck-m an ifo lds ( k s l ) m odelled a fte r re a l Banach sp a c e s Eo and E, su ppose M i s an oriented subm anifold of Y with the in clusion M Ç Y m odelled a fte r som e in clusion F £ E of r e a l Banach sp a c e s , and suppose f; X -» Y i s а С к-Ф .«(Т )-тар t r a n sv e r sa l to M, w here T: E 0 -> E i s som e F red h o lm o p era to r t r a n sv e r sa l to F . Then f _1 (M) is an oriented subm anifold of the T -p u ll-b ack X T of Y by f, with the in clusion m odelled a fte r the in clusion T _:l( F ) £ E o of Banach sp a c e s . The re s tr ic t io n f: f _1(M) -» M off i s a Ck-(T + L c(T_:l (F), F ))-m ap , w here ¥ : T ‘ 1 (F) -» F i s the re str ic t io n of T , which i s a F red h o lm op erato r of the sam e index a s T . (D efinition: C a ll the oriented Ck-m anifold f " 1(M) the T -p u ll-b ack of M by f ) .

(iii) In both c a s e s (i) and (ii) the re g u la r points fo r f a re exactly the re g u la r points fo r f that belong to the se t f ^ M ) , and the re g u la r v a lu e s fo r ? " a r e ex ac tly the re g u la r v a lu e s fo r f that belong to M. F u rth erm o re , in c a se (ii) when T = I and E 0 = E , f and f have the sam e orien tation s a t each of these re g u la r points.

The statem en t of the theorem i s much s im p lified when X and Y a re f in ite dim ensional:

P a r t ic u la r c a se : Suppose f: X -* Y i s any C k-m ap (k s 1) t r a n sv e r sa l to som e subm anifold (or, re sp e c tiv e ly , orien ted subm anifold) M of У, where X and Y a re fin ite-d im en sio n al C k-m an ifo ld s (or, re sp ec tiv e ly , oriented Ck-m an ifo ld s). Then f _:l(M) i s a subm anifold (or, re sp ec tiv e ly , an oriented subm anifold) of X of d im ension dim M + dim X - dim Y, and the re str ic t io n f: f “1 (M) -» M of f i s a Ck -m ap that has a s re g u la r points exactly the re g u la r points of f that a re in the se t f _1(M). F u rth erm o re , if X and Y a re oriented and m odelled a fte r the sam e re a l sp ace , and if dim M / 0, f and f have the orientation a t each of those re g u la r points.

T h eorem 2: An a t la s fo r the in clu sion f _1 (M )£ X of Ck-B an ach -m an ifo ld s in c a se (i), c o n s is ts of a ll C k-d iffeo m ó rp h ism s '1 i s of the type T<p + m ap into som e fin ite-d im en sio n al su b sp ace of F g , fo r a rb itr a ry c h arts |3 m odelling the in clusion M £ Y a fte r the in clusion F g £ E g , and fo r a rb itr a ry T^ e Фч(Еа , E e ) chosen sa tis fy in g RfT^,) + Fe = EB (e .g . = (fifa '1 ) ’ (a(x)). In c a se (ii) wereq u ire the <p to be o r ien ta tio n -p re se rv in g and the to be a ll equal to T: E o -* E , and we obtain what we ca ll the pu ll-b ack a t la s fo r the in clusion f ' 1(M) £ X T of orien ted C k-m an ifo ld s, which i s then m odelled a fte r the in clusion T "1 ( F )£ E q of Banach sp a c e s .

C o ro llary : T akin g M = {y } in the tr a n sv e r sa l ity theorem , we obtain that if у i s a re g u la r value fo r f then f " I (y) i s a subm anifold with dim ension equal to the index of f, and that in c a se (ii) th is subm anifold i s oriented in a uniquely defined way. T h is ju s t if ie s the defin itions, in the introduction, of the d egree of p ro p er F red h o lm m aps with positive index: f ‘ 1 (y) = deg(f, X, y).

T r a n s v e r s a l i t y theorem

3 04 ISNARD

C o ro llary : Sn_1 = {x e Rn | £ x ? = 1} i s an oriented C“-subm an ifo ld of Rn. M ore gen era lly , le t E be any re a l Banach sp ace such that the function x - |x | is a Ck-m ap ( k i 1) on E - { 0 } (e. g. any re a l H ilbert sp ace , with к = +oo). Then the se t { x £ E j [ x I = 1} i s an oriented Ck-subm an ifo ld of E.

P ro o f: 1 is a r e g u la r value fo r the function cp-. x -* |x | E R on E - { 0 } , becau se

• / \ . x+tx - X I I<p' (x) • x = lim J— — - 1— 1— 1 = x t-> о t

Hence the fin ite-d im en sio n al c a se i s se ttled by the p rev iou s co ro lla ry . F o r the in fin ite-d im en sion al c a se , we choose E = E i ® F where dim F = 1, we ca ll P the p ro je c to r onto E i null on F , we choose u f 0 in F , and we app ly the t r a n sv e r sa l ity theorem to the C k-m ap x -» P(x) + |x |u on E - { 0 } , which i s tr a n sv e r sa l to u + E ¡ and has d er iv a tiv e s of the type I + m ap into F (so i s a Ck-(I+ L c (E, E ))-m ap .

An im m ediate consequence of the tr a n sv e r sa l ity theorem i s the

T ra n sv e r sa lity property : Suppose f, X, Y, X and Y a re a s in any of the defin itions of the d egree (section C), and su ppose the re str ic t io n X -» Y of f i s t r a n sv e r sa l to a subm anifold M of Y or, re sp ec tiv e ly , to an oriented subm anifold M of Y. Then the re str ic t io n f: f '^ M ) -* M of f i s a p ro p er map (because f i s so ), and if one co n sid e rs the subm anifold or, re sp ec tiv e ly , the oriented subm anifold f _:L(M) П X of X given by the t r a n sv e r sa l ity theorem , the re s tr ic t io n f-^M ) n x -» M of f”i s а С 1-Ф 0- т а р , or, re sp ec tiv e ly , С1- Ф *- т а р . F u rth e rm o re , any у GM - f(3X) that i s a re g u la r value fo r e ith er f o r f, i s a re g u la r value fo r both those m ap s. We have a lso deg(f, X, y) = deg(f^ f'-^M), y), fo r a ll y e M - f(9X).

The proo f of the th eo rem s in this section is based on the follow ing re su lt, which m ay be re g ard e d a s an in v erse function theorem m odulo F , when one c a l ls a r e g u la r point m odulo F fo r a m ap g: X -► E any x e X such that R (g ’ (x)) + F = E .

T h eorem 3: Suppose g: X -» E i s а С к-Ф о -та р , ( k î 1), where X i s a rea l, com plex, or oriented Ck-B an ach -m an ifo ld , and suppose R (g '(x 0)) + F = E fo r som e x0 e X and som e F lin e a r su b sp ace of E . Then there is som e Ck- d iffeom orph ism cp between som e open su b se t U , of X and som e open su b se t of E , such that (cp-g) (Uv) i s contained in som e fin ite-d im en sio n al lin ear su b sp ace of F , and such that x 0e U , and cp(x0) = g (x0). F u rth erm o re , there i s one such cp that i s o r ien ta tio n -p reserv in g , when X i s an oriented C k- m anifold m odelled a fte r E, g i s а С к-Ф „.-тар , F f 0 and R (g '(x 0)) + F = E.

P ro o f: T ak in g com po sition s with ch arts a t x, we m ay suppose that X i s an open su b se t of som e Banach sp ace Eo (which i s equal to E the oriented c a se ) . By lem m a 3 there i s som e L e L (E q, F ) such that g'(xo) + L i s an i s o m orph ism Е о = E, which i s o rien ta tio n -p re se rv in g in the oriented c a se .Then x 0 i s a re g u la r point fo r g + L - L (xo), with p o sitive orientation in the oriented c a se . We apply to g + L - L (x 0) the in v erse m apping theorem (section C), obtaining U^,, and cp by re str ic tio n .

F o r m ap s of non -zero index we need the follow ing gen eralization :

T h eorem 4: Suppose g: X E i s а Ск-Фч- т а р , where q i s any in teger, к ё 1, X i s a rea l, com plex or oriented Ck-B an ach -m an ifo ld , and suppose

IAEA- SMR-11/36 305

R (g'(xo)) + F = E fo r som e x 0 e X and som e F lin e a r su b sp ace of E . Suppose T e ФЧ(Е 0, E) i s such that R(T) + F = E where E 0 i s an a r b it r a r y Banach sp a c e . Then there i s som e Ck-d iffeom orph ism tp between som e open su b se t Uv of X and som e open su b se t of E 0 such that (Ttp-g) (U ) i s contained in som e fin ite-d im en sio n al lin ea r su b sp ace of F , with xo e ü , . F u rth erm o re , there i s one such tp that i s o rien ta tio n -p reserv in g , when X i s an oriented Ck-m anifo ld m odelled a fte r E, g i s а Ск- Ф *(Т )-т а р , T "1 (F) f 0, and R (g'(x)) + F = 0.

P ro o f: B y lem m a 1, there i s som e T e Ф_Ч(Е, E 0) such that I - T T and I - T Ta re p ro je c to r s onto fin ite-d im en sio n al su b sp a c e s of F or, re sp ec tiv e ly ,T _1F , and such that R (T g)'(x) + T _:LF = E 0. B y lem m a 2 f g is а СкФ *(1 )-тар (because when S e Ф*(Т), then TS e ® *(fT ) = Ф*(1)). We now apply theorem 3 to í g and obtain <p such that (tp - Tg) (Uv ) i s contained in a fin ite-d im en sio n al su b sp ace of T _1 F . But then (Tip- TTg) (Ц Д and hence (Ttp-g) (U,,), a re contained in fin ite-d im en sio n al su b sp a c e s of F (because R ÍT Í- I ) £ F ) .

Definition: When T e L (E 0, E) and F £ E we c a ll T + L C(E 0, F) = { S £ L (E o , E) |I S -T i s com pact and R (S-T ) £ F } .

P ro o f of the pu ll-b ack theorem : T ake M = Y and F = E j in the follow ing proof of c a se (ii) (the oriented ca se ) of the tr a n sv e r sa l ity theorem .

P ro o f of the t r a n sv e r sa l i ty theorem and of theorem 2: B y theorem 4, the dom ains U,, of the <p in theorem 2 co v er f'-^M ). U se now:

L em m a 4: In the conditions of theorem 4, we have tp(Uv n g '1 (F)) == <P(U,р) n T _1 F , and if we c a ll h = grp'1, then (h-T) (U^) is contained in a fin ite -d im en sio n al su b sp ace of F . A lso , h i s a Ck-(T^t- L c(Eo, F ))-m ap , and h as a re s tr ic t io n h: T _:L(F) П tpCU ) -► F which i s a C k-(T + L c(T_:lF , F ))-m ap , w here T: T _1 (F) -* F i s the re s tr ic t io n of T . Hence h and h a re C k-®q-m ap s. The re g u la r points fo r h a re exactly the re g u la r points fo r h that a re in T _1(F) n Uv . F u rth erm o re , in the oriented c a se when T = I, E = Eo, F f 0 and g i s а Ск- Ф *- т а р , the orientation fo r g of any re g u la r point u 6 f _1 (M) n U<p co in cides with the orientation of <p(u) a s a re g u la r point fo r both h and h.

P ro o f: F o r x e U ,, we have tp(x) e T _:LF -» T(<p(x)) E F M g(x) £ F . We have a lso (h-T) (<p(U^)) = ((g 1 - T)<p)(U^) = (g-T<p) ( U , ) E F „ a fin ite-d im en sio n alsu b sp ace of F . F o r w e (Uv ) n T _1 (F). Then (h-T) (w) e F v and T(w) £ F , so h(w) e F .ÎT'(w) i s the re str ic t io n T ' 1 F -> F of h'(w), hence R(h'(w) - T) £ R (h'(w )-T) £.F^ fin ite-d im en sio n al с F . R(h'(w)) = h'(w) (T _:lF ) = h'(w) ((h'(w))"1 (F)) == R(h'(w)) n F , so

F = F = R(h'(w)) + F _ ER(ÎT'(w)) R ( h '( w ) ) n F ~ R(h'(w)) ~ R ( h ’ (w))

Hence w i s a re g u la r point fo r h if and only if it i s a re g u la r point fo r h.Suppose now T = I, Eq = E , F / 0 and g i s а Ск-Ф+- т а р . The <p provided

by theorem 4 i s then o rien ta tio n -p reserv in g . If u e U ^ n g_1 (M) a re g u la r point fo r g, then by the m ultip lication law of orien tation s (section B) <p(u)

306 ISNARD

i s a re g u la r point fo r h with the sam e orientation , b ecau se h i s a Ck-(I + L C(E, E ))-m ap . We have fo r w = tp(u) R(h'(w)-I) c R(h(w)-I) Ç.F,, fin ite-d im en sio n al Q F , hence the determ in an ts of ÎT'(w): F -» F and h'(w): Е -> E coincide with the determ inant of the com m on re str ic tio n F^, -* F,, . T h ere fo re the orien tation s a lso coincide.

Continuation of p roof of t r a n sv e rsa lity theorem : T o com plete the proof, it su ffic e s to show that in c a se (ii) the ch arts <p m ake f _1 (M) an oriented m anifold.

P ro o f of the p u ll-b ack theorem : C a ll CT the collection of a ll orientation p re se rv in g Ck-d iffeo m o rp h ism s between open su b se ts of X and of E , such that f <p_1 i s a Ck-(T + L C(E 0, E ))-m ap . By what w as a lre ad y proved CT contains an a t la s fo r the topological sp ace X, and it su ffice s to show that CT is an orientation a t la s . L e t x be in the dom ains of q>i and <p2 e CT, ca ll H = (<p2 <P\) '(<Pi(x)), it su ffice s to show that H e G L j( E ) , Since срг and ip2 a re o r ien ta tio n -p re se rv in g C k-d iffeom orp h ism s, it follow s from the m u ltip lication law of o rien tation s that it i s enough to show that H e I + L C(E, E ). L e t (? be a ch art in som e a t la s fo r Y such that f(x) e Ug, ca ll, fo r i = 1, 2,S¡ = (/3 f ip~l) '(<Pi(x)). Then S i and S2 a re in T + L c(Eq, E) and S2 H = S j . L em m a 1 g iv es us som e T £ L (E , E 0) such that Í T - I and Т Т - I a re com pact, so TS-l and T S 2 a re in I + L C(E, E ). Hence T S } = T S 2H i s both in I + L C(E, E) and in H + L C(E, E ), so H £ I + L C(E, E ). Thus C j i s an orientation a t la s .

F in a lly , we o b serv e that if Х -l i s any oriented C k-m anifold obtained attaching to the topo logical sp ace X som e orientation a t la s , then X j s a t i s f ie s the conditions in the definition of the (T )-p u ll-b ack of Y by f if and only if each ch art in a t la s i s in CT (therefore, if and only if X j = X T): In fact, the identity X = X] i s o rien ta tio n -p reserv in g if and only if each ch art in the a t la s fo r Xj is o rien ta tio n -p re se rv in g when it s dom ain i s oriented a s a su b se t of X.

P ro o f that f -1(M) i s oriented: Suppose that u e f " 1 (M) n n u , / 0 , where each O if - T<pj) (U<p¡) c .fin ite-d im en sion al su b sp ace of F,, , where and 02 a re lo ca l ch arts fo r the in clusion M Ç Y . We c a ll H = (<p2 «Pi1)'(<Pi(u)), we know that H 6 G L j(E ) b ecau se <p\ and <p2 a r e ch arts in an a t la s fo r the (T )-pu ll-back of Y by f. We know H h as a re s tr ic t io n Hr : T _1 (F) -* T "1 (F), and we m ust prove Hr e G L ; ( T _1F ) . We c a ll G = (¡32 fo1 ) '(0 i (f(u))), we know G e G L ^ E i ) and G has a re str ic t io n Gr e G L J(F ) . F o r i = 1, 2, we ca ll Sj = (Pi f tp'i) '( ‘Pifu)), we have R (S ¡- T )£ F b e cau se by lem m a 4 f i s a Ck-(T + L c(E 0,F ) ) - m a p . We have G Sj. = (j32 f «Px1) '(<Pi (u)) = S2 H. F ro m R ( S ¡ - T ) £ F we know that S jfT ' 1 (F)) ç F and that S¡ and T induce the sam e lin e a r m ap в: E 0/ T _1F - E / F . C allin g Hq: E q / T '^ F ) = E o / T '^ F ) and Gq : E /F a E / F the iso m o rp h ism s induced by H, re sp ec tiv e ly ,G , we get Gq в = в Hq (from G S j = S2 H). We ob serv e that в i s an iso m o rp h ism E 0/ T _1 (F) = E / F (with in v e rse induced by any T e L (E , Eo), provided by lem m a 1, such that I - T T and Т Т - I has range in side F , or, re sp ec tiv e ly , T _1 (F)). Then H^ = 0_1o Gqo 6. The next two lem m as (5 and 6) te ll us that Gr £ GLc (F) ~ Gq 6 G L j( E /F ) Hq e G L¿ (E q /T ' 1 (F)) Hr e G L J (T ‘ 1 (F)), which com pletes the proof.

L em m a 5: Say 0 F Í E a re r e a l Banach sp a c e s and A = {S £ L (E , Е) |I S(F ) ç F } . Then A is a Banach su b sp ace of L (E , E) (actually, a su b a lgeb ra ), and in A a re defined continuous l in e a r m aps S -* Sr e L (F , F ) and

IAEA-SMR-11/36 307

S - S qG L ( E /F , E /F ) , where S r and S q a re the m aps induced by S. If S-I i s com pact, so a re S r-I and Sq-I, and then S i s an o rien ta tio n -p reserv in g , or, re sp e c tiv e ly , re v e r s in g , iso m o rp h ism E = E if and only if Sr : F = F and Sq : E / F = E / F a re iso m o rp h ism s with the sam e or, re sp ec tiv e ly , with d ifferen t o rien tation s. In p a rticu la r , if S - I has fin ite-d im en sio n al ran ge, so have Sr - I and Sq - I, and then det(S) = det(Sq) • det(Sr).

R em ark : If S ^ I + L C(E, E) we m ay have iso m o rp h ism S : E = E such that S(F ) £ F : F o r in stan ce, in E = i 2 ® ^ 2» define S (x i , x 2, . . . ), (y i, y 2, • • • )) == ((y1# x x, x 2, . . . ), (y2, y3 , . . . )), then S (£ 2 ® 0) t- ® 0.

L em m a 6: L e t 6: E j E 2 be an iso m o rp h ism of re a l Banach sp a c e s , then fo r S e L (E 2, E 2) we have S G G L ¡t(E 2) ~ » 0' 1 o S °в G G L J ( E X).

P ro o f of lem m a 5 :; We have |sr | S j S | and |S q| S |s|, hence S - Sr and S -> Sq a re continuous on A. If S i s com pact, so a re Sr and S q, hence if S - I i s com pact so a r e S r - I and S q - I. Suppose fro m now on that S - I , S r - I and S q - I a re com pact. Then S £ G L(E) [ S r e G L(F) and S q £ G L (E /F ) ], b ecau se if S G G L(E) then Sr i s in jective and Sq i s su r je c tiv e , hence, sin ce S t e $ 0(F, F ) and Sq e ®0(E /F , E / F ) , we have S r e G L(E) and S qe G L (E /F ) .By ch apter III, sectio n B, c o ro lla ry 4, each path-com ponent of GLC (E) ПА contains one path-com ponent of G L f(E) n A . Then ev ery S G G LC (E) П A is connected by som e continuous path t -» S t £ GLC(E) n A to som e elem ent of GLf (E) n A. The paths t -> (St)r g G L C(F) and t -» (St )q g GLC(E /F ) a re a lso continuous, and the orien tation s a re constant along those paths, so it su ff ic e s to co n sid er the orien tation s when S GGLf(E) n A. But then R(S-I) £ F q fin ite-d im en sio n al Q E, R(St -I) £ F n F q , and

R (Sa-I) ç ^ : +F Fo4 ; ~ F F 0 n F

So to p rove det(S) = det(Sr) • det(Sq) it su ffice s to con sid er the c a se where E is fin ite-d im en sio n al. Then E = F x (E /F ) , so it d e r iv e s from :

L em m a 7: If E i and E 2 a re fin ite-d im en sio n al v ec to r sp a c e s and S £ L ( E i x E 2, E i x E 2) i s defined by S(u 1, u2) = (Siu 1 + Au2 , S2 u2), fo r S ^ L f E j , E J , S 2 e L (E 2j E 2) and A e L (E 2, E j ) , then det(S) = d e t ^ ) • det(S2 ).

P ro o f: S = S2§ i , where S (u i, u 2) = (S iu i + Au2, u2), and S2 (щ , u 2) = (u i, S 2 (u2)). Then det(S) = det(S2 ) • d e tiS j ) = d e t(S 2) • det(S j ), b ecau se RfSx -I) £ E 2 x 0 and R(S2 - I ) £ 0 x E 2.

P ro o f of lem m a 6 : S -> в'1 S в i s an iso m o rp h ism L ( E 2, E 2) = L (E i , E i ) thathas fo r re s tr ic t io n an hom eom orphism G L C(E 2) = G L jfE j) , which then tak es path-com ponent onto path-com ponent.

L em m a 1: If T e L (E 0 , E) and R(T) + F = E , then T G Ф ^Ео, E) *-* the r e s t r i c tion T " 1 (F) -♦ F of T i s in $ q(T '1 F , F ) . F o r each in teger q the se t { T e $ q(E o ,E ) I R(T) + F = E } is open in L (E o , E ). A lso , fo r each T in this se t R(T) i s c lo sed , E = R(T) © F 0 fo r som e fin ite-d im en sio n al F o £ F , and there i s som e Í g Ф.Ч(Е, E 0) such that I - T T and I - Í T a re p ro je c to r s onto fin ite-d im en sio n al su b sp a c e s of F , or, re sp ec tiv e ly , of T ' 1 (F), and such

3 0 8 ISNARD

t h a t { S e < M E 0, E ) I R ( S ) + F = E } = { S £ L ( E 0, E ) | T S e Ф 0( Е 0 , E 0) a n d R ( T S ) + T ' ^ F = E o } . (In p a r t i c u l a r , t a k i n g F = E i n th e s t a t e m e n t , w e g e t t h a t t h e s e t s $ q( E 0 , E ) a r e o p e n in L ( E , E ) , a n d t h a t i f T e Ф ч( Е о , E ) th e n R ( T ) i s c l o s e d , E = R ( T ) ® F o f o r s o m e f i n i t e - d i m e n s i o n a l F o £ E , a n d t h e r e i s s o m e f e Ф - Ч( Е , E 0 ) s u c h t h a t R ( T T - I) a n d R ( T T - I) a r e f i n i t e - d i m e n s i o n a l . A l s o , i f T e L ( E 0 , E ) a n d E = R ( T ) + F ^ w h e r e F j i s a f i n i t e - d i m e n s i o n a l l i n e a r s u b s p a c e o f E , t h e n T € Ф Ч( Е 0, E ) q = d i m T " 1( F 1 ) - d i m F j < + 00) .

P r o o f : S u p p o s e T e L ( E 0, E ) a n d R ( T ) + F = E . T h e r e s t r i c t i o n T ' ^ F ) - F i s in ФЧ( Т _1 ( Р ) , F ) i f a n d o n l y i f T £ Ф Ч( Е 0, E ) b e c a u s e b o t h n u l l - s p a c e s a r e N ( T ) , a n d

F F _ R ( T ) + Fd i m T ( T - i F ) d i m R ( T ) n F - d i m R ( T )

S u p p o s e n o w T e ФЧ( Е 0 , E ) a n d R ( T ) + F = E . T h e f i n i t e - d i m e n s i o n a l v e c t o r s p a c e E / R ( T ) h a s s o m e b a s i s c o n s i s t i n g o f th e q u o t i e n t c l a s s e s o f f i n i t e l y m a n y e l e m e n t s o f F , a n d w e c a l l F 0 t h e i r l i n e a r s p a n . T h e n F 0 £ F , d i m F 0 < +co, a n d E = R ( T ) Д, F 0 , w h e r e f o r th e m o m e n t t h i s d i r e c t s u m i s in t h e a l g e b r a i c s e n s e o n ly , 1. e . w e h a v e n o t y e t p r o v e d t h a t th e c o r r e s p o n d i n g p r o j e c t o r s a r e c o n t i n u o u s . F r o m d i m N ( T ) < +00 w e g e t [ 2 0 ] t h a t Eo = Е ^ Щ Т ) f o r s o m e E i w h i c h i s a c l o s e d s u b s p a c e o f E o , b e c a u s e i t i s t h e n u l l - s p a c e o f a c o n t i n u o u s l i n e a r p r o j e c t o r . T h e n E j i s a B a n a c h s p a c e . W e o b s e r v e t h a t th e m a p S : E ! x F o -* E d e f i n e d b y S ( u , v ) = T u + v i s a n i s o m o r p h i s m , b e c a u s e b y t h e o p e n m a p p i n g t h e o r e m [ 2 0 ] a n y c o n t i n u o u s l i n e a r b i j e c t i o n b e t w e e n B a n a c h s p a c e s h a s a c o n t i n u o u s i n v e r s e . T h e n E = R ( T ) ® F 0 i s a d i r e c t s u m i n t h e t o p o l o g i c a l s e n s e , b e c a u s e i t i s th e i s o m o r p h i c i m a g e ( b y S ) o f a d i r e c t s u m . R ( T ) i s c l o s e d b e c a u s e i t i s t h e n u l l - s p a c e o f s o m e c o n t i n u o u s l i n e a r p r o j e c t o r .

D e f i n e T : E E 0 t o b e n u l l o n F o a n d to b e o n R ( T ) t h e i n v e r s e o f th e r e s t r i c t i o n E x = R ( T ) o f T , t h e n T T a n d T T a r e p r o j e c t o r s o n t o E j o r , r e s p e c t i v e l y , R ( T ) n u l l o n N ( T ) o r , r e s p e c t i v e l y , F o . H e n c e , I - T T a n dI - T T a r e p r o j e c t o r s o n t o N ( T ) o r , r e s p e c t i v e l y , F 0 .

S u p p o s e , n o w , t h a t S £ L ( E 0 , E ) a n d E = R ( S ) + F , t h e n R ( T ) = R ( f S ) + Í F , s o E = R ( f S ) + f ( F ) + T " 1 ( F ) = R C f S ) + T ' 1 ( F ) . C o n v e r s e l y , i f S < E L ( E 0, E ) i s s u c h t h a t E 0 = R f f S ) + T _1 ( F ) , t h e n R ( T ) = R ( T T S ) + T ( T _1( F ) ) , s o E = R ( T T S ) + T ( T _1 ( F ) ) + F = R ( T T S ) + F = R ( S ) + F , b e c a u s e R ( S - T T S) С £ R ( I - T T ) Ç F .

F o r a n y S e L ( E o , E ) w e h a v e S £ Ф Ч( Е 0 , E ) <-* T S £ Ф о ( Е 0 , E o ) b e c a u s e o f

L e m m a 2 : S u p p o s e S € L ( E 0 , E j ) a n d T e L ( E j , E 2 ), w h e r e E 0 , E j a n d E 2 a r e r e a l o r c o m p l e x B a n a c h s p a c e s . I f a n y t w o o f t h e o p e r a t o r s S , T , a n d T S a r e F r e d h o l m o p e r a t o r s , t h e n s o a r e a l l t h r e e , a n d i n d e x (T S ) == i n d e x ( T ) + i n d e x S .

P r o o f : In R e f . [ 2 0 ] .T o c o m p l e t e th e p r o o f o f l e m m a 1 i t r e m a i n s t o b e s h o w n t h a t t h e s e t

{ S e L ( E 0 , E ) I Í S e Ф 0( Е 0 , E 0 ), R ( T S ) + T ’ ^ F ) = E 0 } i s o p e n in L ( E 0 , E ) . F r o m th e c o n t i n u i t y o f t h e t r a n s f o r m a t i o n S -> T S i t s u f f i c e s to p r o v e t h a t th e s e t { V £ Ф0( Е 0 , E o ) | R ( V ) + T _:l( F ) = E 0 } i s o p e n i n L ( E 0 , E 0 ). T h i s i s p r o v e d in

IAEA- SM R-11/36 3 0 9

L e m m a 3: I f E o , E a r e r e a l o r c o m p l e x B a n a c h s p a c e s , a n d F i s a l i n e a r s u b s p a c e o f E t h e n t h e s e t { T e ' $ 0(E0, E ) | R ( T ) + F = E } i s o p e n a n d e q u a l t o t h e s e t { H - L | H : Ец s E i s o m o r p h i s m , L e L f ( E o , F ) } w h e r e L f ( E o , F ) == { L £ L ( E 0, E ) j R ( L ) £ F , d i m R ( L ) < + o o } . A l s o , i f E i s a r e a l B a n a c h s p a c e , T G Ф*(1) £ L ( E , E ) , a n d R ( T ) + F = E f o r s o m e F / 0 l i n e a r s u b s p a c e o f E , the.n t h e r e i s s o m e L £ L ( E , E ) w i t h f i n i t e - d i m e n s i o n a l r a n g e c o n t a i n e d i n F , s u c h t h a t H = T + L : E = E i s a n o r i e n t a t i o n - p r e s e r v i n g i s o m o r p h i s m .

P r o o f : T h e s e t o f t h e H - L a s a b o v e i s o p e n i n L ( E o , E ) b e c a u s e i t i s o p e n f o r e a c h f i x e d L s i n c e t h e s e t o f a l l i s o m o r p h i s m s E o 3 E i s o p e n in L ( E o , E ) ( b e c a u s e G L ( E ) i s o p e n i n L ( E , E ) [ 2 0 ] ) .

E a c h H - L a s a b o v e i s i n Ф 0(Е о. E ) b e c a u s e L i s c o m p a c t . A l s o R ( H - L ) + F = R (H ) + F = E b e c a u s e R ( L ) £ F .

S u p p o s e n o w T e Ф ц (Е о , E ) i s g i v e n s u c h t h a t E = R ( T ) + F . T h e n b y l e m m a 1, E = R ( T ) ® F 0 f o r s o m e F 0 £ F . F r o m i n d e x ( T ) = 0 w e g e t d i m N ( T ) = d i m F 0 , s o t h e r e i s s o m e i s o m o r p h i s m N ( T ) = F 0 t h a t w e e x t e n d t o s o m e L e L ( E 0 , E ) , w h i c h w e d e f i n e t o b e n u l l on s o m e E j c h o s e n s o t h a t E 0 = E-l ® N ( T ) . T h e n T + L i s a n i s o m o r p h i s m E 0 = E , b e c a u s e i t s r e s t r i c t i o n s E x = R ( T ) a n d N ( T ) = F 0 a r e s o .

S u p p o s e T 6 Ф ,(1 ) £ L ( E , E ) a n d E = R ( T ) + F , F / 0 . B y w h a t w a s j u s t p r o v e d w e h a v e t h e n T + L e G L ( E ) f o r s o m e L w i t h f i n i t e - d i m e n s i o n a l r a n g e c o n t a i n e d in F . T h e n T + L G G L ( E ) n Ф *(1), a n d w e m a y s u p p o s e T + L i s o r i e n t a t i o n - p r e s e r v i n g , o t h e r w i s e w e r e p l a c e T + L b y ( I+ C ) ( T + L ) = T + L j , w h e r e С e L ( E , E ) i s s u c h t h a t I + C e G L ¿ ( E ) a n d R ( C ) £ F i s f i n i t e - d i m e n s i o n a l : T h e n L x = L + C ( T + L ) h a s r a n g e i n R ( L ) + R ( C ) , w h i c h i s f i n i t e - d i m e n s i o n a l £ F . S u c h С m a y b e o b t a i n e d b y c h o o s i n g s o m e o n e - d i m e n s i o n a l F £ F , d e f i n i n g J = I + С t o b e - I o n F a n d t o b e I on s o m e E s u c h t h a t E = E © F , th e n R ( C ) = R ( J - I ) = F , a n d th e d e t e r m i n a n t o f J i s t h e d e t e r m i n a n t o f - I : F -> F , w h i c h i s - 1 .

F . H O M O T O P Y P R O P E R T Y

T h r o u g h o u t t h i s s e c t i o n J a n d J a r e s u b s e t s o f t h e s c a l a r f i e l d (R o r C ) , w i t h J o p e n a n d J 2. J , a n d X , X , Y, Ÿ a r e a s in th e d e f i n i t i o n s o f t h e d e g r e e .W e h a v e , t h e r e f o r e , t h r e e c a s e s : th e c o m p l e x c a s e , t h e o r i e n t e d c a s e , a n d th e g e n e r a l c a s e o f t h e m o d 2 d e g r e e . X x J , o r , r e s p e c t i v e l y , Y x J i s th e p r o d u c t m a n i f o l d , w h i c h i s c o m p l e x , o r o r i e n t e d , o r r e a l i f X o r , r e s p e c t i v e l y ,Y i s s o , a n a t l a s f o r X x J i s { o x I | a e a t l a s f o r X , I = i d e n t i t y o n J } . Y x J i s d e f i n e d s i m i l a r l y . F r o m l e m m a 1 f o l l o w i n g i t f o l l o w s t h a t X x J i s o r i e n t e d i f X i s s o .

H o m o t o p y p r o p e r t y 1 : S u p p o s e (x , t) -* f t (x) i s a c o n t i n u o u s p r o p e r m a p X x J t e x t e n d i n g s o m e C a r n a p X x J - * Y w h i c h i s s u c h t h a t f o r e a c h t e J th e r e s t r i c t i o n f t : X - * Y i s а С 1- Ф 0- т а р ( o r , r e s p e c t i v e l y , а С 1- Ф * - т а р , f o r t h e o r i e n t e d c a s e ) . L e t у b e i n Y - f t ( 3 X ) f o r a l l t i n s o m e c o n n e c t e d s u b s e t С o f J . T h e n d e g ( f t , X , y) i s c o n s t a n t f o r t e C .

A m a p (x , t) -* f t (x) a s a b o v e i s c a l l e d a p r o p e r C 1- h o m o t o p y o f Фо- m a p s . T h e t h e o r e m h a s a m o r e g e n e r a l f o r m w h e r e b o t h у a n d t h e d o m a i n o f f a r e p e r m i t t e d t o c h a n g e w i t h t:

H o m o t o p y p r o p e r t y 2 : S u p p o s e Г2 a n d Г2 a r e s u b s e t s o f X x s c a l a r f i e l d , w i t h Г2 o p e n in X x s c a l a r f i e l d a n d c o n t a i n e d i n ñ . S u p p o s e (x , t) -*■ f t (x) i s

310 ISNARD

a c o n t i n u o u s p r o p e r m a p f t -* Ÿ t h a t e x t e n d s s o m e C 1- m a p Г2 - Y s u c h t h a t f o r e a c h t s c a l a r , i f G t = { x € X | (x , t) £ i s n o t e m p t y , t h e n f t : G t -> Y i s а С 1- Ф о _т а р ( o r , r e s p e c t i v e l y , а С 1- Ф * - т а р , in th e o r i e n t e d c a s e ) . S u p p o s e t • * y t £ Y i s a c o n t i n u o u s m a p o n s o m e c o n n e c t e d s u b s e t С o f th e s c a l a r f i e l d s u c h t h a t y t f f t (x) i f (x, t) £ f t - f 2. T h e n d e g ( f t , G t , y t ) i s c o n s t a n t f o r t i n C . I f G t = 0 f o r s o m e t in C , t h e n d e g ( f t , G t, y t ) = 0 f o r a l l t in C .

P r o o f : T h e m a p q>: f t - * Ÿ x s c a l a r f i e l d d e f i n e d b y <p(x, t) = (ft (x ) , t) i s c o n t i n u o u s a n d p r o p e r , b e c a u s e i f К i s a c o m p a c t s u b s e t o f Y x s c a l a r f i e l d , t h e n <p_1(K ) i s a c l o s e d s u b s e t o f th e c o m p a c t s e t { ( x , t) £ f t | f t (x) £ t t j K } , w h e r e (y, t) = y e Y w h e n t i s s c a l a r . F r o m t h e n e x t l e m m a , w e o b t a i n t h a t t h e r e s t r i c t i o n П -» Y x s c a l a r f i e l d o f tp i s а С ^ Ф о - т а р ( а С 1- Ф + - т а р in t h e o r i e n t e d c a s e ) s u c h t h a t i f (y, t) £ ( Y x J ) - (p (Û -C l ), a n d y i s a r e g u l a r v a l u e f o r f t , t h e n (y, t) i s a r e g u l a r v a l u e f o r <p s u c h t h a t deg(<p, (y, t)) == d e g ( f t , G t, y ) .

L e m m a 1: S u p p o s e E , E i a n d F a r e r e a l o r c o m p l e x B a n a c h s p a c e s a n d T £ L ( E x F , E j x F ) i s o f th e t y p e T ( u , v ) = ( Î u + S v , v ) , w h e r e T e L ( E , E x ) a n d S £ L ( F , E -l) . T h e n , f o r a n y i n t e g e r p w e h a v e :

(i) T e i p( E x F , E i x F ) — T е ф р( Е , E i ) .( i i ) T i s a n i s o m o r p h i s m E x F s E i x F i s a n d o n l y i f T i s a n i s o m o r p h i s m E = E i .W h e n th e B a n a c h s p a c e s a r e r e a l , g i v e n s o m e A £ Фр ( Е , E x ) w e h a v e( i i i ) T 6 Ф * ( А x I ) « - * T £ Ф * (A)( w h e r e ( A x l ) ( u , v ) = (A u , v ) , u e E , v £ F ) .( iv ) W h e n t h e s p a c e s a r e r e a l , E = E i , t h e n T £ Ф*(1) n G L ( E x F ) *-* T £ Ф * (1)П n G L ( E ) , a n d t h e n T a n d T h a v e t h e s a m e o r i e n t a t i o n .

P r o o f : (i) a n d ( i i ) a r e p r o v e d in s e c t i o n B . T o p r o v e ( i i i ) , o b s e r v e t h a t f r o m(i) i t f o l l o w s t h a t f o r 0 S t S 1 t T + ( 1 - t ) ( A x l ) i s a F r e d h o l m o p e r a t o r E x F <-> E i x F i f a n d o n l y i f t T + (1 - t ) A i s a F r e d h o l m o p e r a t o r E -» E i . T o p r o v e th e o r i e n t a t i o n s in ( iv ) , w e g e t f r o m s e c t i o n В a c o n t i n u o u s p a t h t -* T t £ Ф*(1) n G L ( E ) c o n n e c t i n g T t o s o m e e l e m e n t T x o f G L f (E ) (0 fi t S 1 ) . W e d e f i n e T t (u , v) = ( T t u + t S v , v ) , w h i c h g i v e s a c o n t i n u o u s p a t h t - * T t £ G L ( E x F ) П Ф^.(1) c o n n e c t i n g T t o T j x I , w h i c h h a s t h e s a m e d e t e r m i n a n t , h e n c e th e s a m e o r i e n t a t i o n a s T i . S i n c e th e o r i e n t a t i o n s a r e c o n s t a n t a l o n g t h o s e p a t h s , i t f o l l o w s t h a t T a n d T h a v e t h e s a m e o r i e n t a t i o n .

C o n t i n u a t i o n o f p r o o f o f h o m o t o p y p r o p e r t y : F o r a l l t 6 С w e h a v e t h e n deg(<p, Г2, ( y t, t)) = d e g ( f t, G t , yt ) (w h e n y t i s n o t a r e g u l a r v a l u e , w e g e t t h e r e s u l t m a k i n g y n -> yt> y n r e g u l a r v a l u e f o r f t , t f i x e d ) . B u t s i n c e { ( y t » * ) I t £ C } i s a c o n n e c t e d s u b s e t o f Y x J - < p ( f t -Q ) , i t f o l l o w s t h a t d e g ( f t (yt , t)) i s c o n s t a n t f o r t in C , b y th e i n v a r i a n c e p r o p e r t y .

G . T H E M U L T I P L I C A T I V E P R O P E R T Y

S u p p o s e f: X -» Y a n d g : Y -♦ W a r e c o n t i n u o u s p r o p e r m a p s t h a t h a v e f o r r e s t r i c t i o n s С ^ Ф о - т а р ( o r , r e s p e c t i v e l y , С 1- Ф %- т а р з in t h e o r i e n t e d c a s e ) X -» Y a n d Y -* W, w h e r e X , Y a n d W a r e r e a l o r c o m p l e x C 1- B a n a c h - m a n i f o l d s ( o r , r e s p e c t i v e l y o r i e n t e d C 1- m a n i f o l d s m o d e l l e d a f t e r s o m e r e a l

IAEA-SM R-11/36 311

B a n a c h s p a c e E ) , t h a t a r e o p e n s u b s e t s o f a r b i t r a r y t o p o l o g i c a l s p a c e s X , Ÿ a n d W . W e s u p p o s e a l s o t h a t X a n d Y a r e H a u s d o r f f s p a c e s . L e t { Yj I j } b e t h e c o l l e c t i o n o f c o n n e c t e d c o m p o n e n t s o f Y - f ( 9 X ) , a n d l e t e a c h y¡ e Yj b e a n a r b i t r a r y p o i n t . T h e n d e g ( g • f, X , w) = Ç d e g ( f , X , yj )• d e g ( g , Yj, w ), f o r e v e r y w e W - g ( f ( 9 X ) U 9 Y ) . A n d t h i s s u m i s f i n i t e b e c a u s e w e h a v e d e g ( g , Yj, w) = 0 f o r a l l b u t f i n i t e l y m a n y i n d i c e s j . R e m a r k : g f i s С ^ - Ф о - т а р f r o m X t o W , b y s e c t i o n E , l e m m a 2 , b u t , i n t h e o r i e n t e d c a s e , i t m a y f a i l t o b e а С^-Ф^-тар, b e c a u s e t h e c o m p o s i t i o n o f t w o o p e r a t o r s b e l o n g i n g to I * (I) m a y b e o u t s i d e Ф*(1). In t h i s c a s e w e d e f i n e t h e d e g r e e o f g f o n X a s th e o n e w h e n X i s r e p l a c e d b y X j = t h e (I) - p u l l - b a c k o f Y b y f ( s e c t i o n E ) , w h a t d o e s n o t c h a n g e d e g ( f , X , y j ) f o r e a c h j a n d w h a t m a k e s g f а С 1-Ф*-тар, b e c a u s e f b e c o m e s a C M l + L C( E , E ) ) - m a p . S o , i n t h e o r i e n t e d c a s e , w e s h a l l s u p p o s e i n t h e p r o o f t h a t X h a s a l r e a d y b e e n r e p l a c e d , s o t h a t f i s a C M l + L C ( E , E ) ) - m a p .

P r o o f : g _1 (w) i s a c o m p a c t s u b s e t o f Y - f ( 9 X ) = y Y j , s o g " 1 (w) i s c o n t a i n e d i n a t m o s t f i n i t e l y m a n y o f t h e Y j . F o r t h e r e m a i n i n g Y j w e h a v e g ' 1(w) n n Y j = j), h e n c e d e g ( g , Y j , w) = 0, b y th e s u r j e c t i v i t y p r o p e r t y , (g f ) '1 (w) == f ' M g ’ M w )) Q y f ' M Y j ) , s o d e g f g f , X , w) = S d e g ( g f , f _1( Y j ), w ) , b y th e a d d i t i v i t y p r o p e r t y . A l s o , f o r e a c h j d e g ( f , J X , y j ) = d e g ( f , f _1( Y j ) , y j ) , b y t h e e x c i s i o n p r o p e r t y . S o i t s u f f i c e s to p r o v e f o r e a c h j t h a t d e g ( g f , f ‘ 1( Y j ) , w) == d e g ( f , f _1( Y j ), y j ) • d e g ( g , Y j , w ) . In o t h e r t e r m s , w e m a y s u p p o s e Y = Y i s c o n n e c t e d a n d X = f _1 (Y) = X . T h e n к = d e g ( f , X , y) i s c o n s t a n t f o r у G Y , a n d a l l w e n e e d t o s h o w i s t h a t d e g ( g f , X , w) = к d e g ( g , Y , w ) . W e m a y s u p p o s e w i s a r e g u l a r v a l u e f o r g , b y t h e i n v a r i a n c e p r o p e r t y . T h e n g _1(w) i s a f i n i t e s e t . T h e c a s e g ‘ 1 (w) i s e m p t y i s s e t t l e d b y t h e s u r j e c t i v i t y p r o p e r t y .S u p p o s e , n o w , g _1 (w) = { у ц , . . . , y q } . E a c h y ¡ i s a r e g u l a r v a l u e f o r g, h e n c e i t i t s o m e o p e n s e t Vi w h e r e g i s a C 1- d i f f e o m o r p h i s m o n t o s o m e o p e n s e t c o n t a i n i n g w . W e m a y s u p p o s e t h a t th e V¡ a r e p a i r w i s e d i s j o i n t . B y th e a d d i t i v i t y p r o p e r t y i t s u f f i c e s t o s h o w t h a t f o r e a c h i d e g ( g f , f " 1 ^ ), w) == к d e g ( g , V j , w ) . S o , r e p l a c i n g X a n d Y , w e m a y s u p p o s e t h a t g i s a C 1- d i f f e o m o r p h i s m , a n d i t s u f f i c e s t o s h o w t h a t d e g ( g f , X , g (y ) ) == d e g ( f , X , y) • d e g ( g , Y , g ( y ) ) f o r a l l y e Y . W e m a y s u p p o s e y i s a r e g u l a r v a l u e f o r f . T h e n f _1(y) i s a f i n i t e s e t { x j , . . . , x p} = ( g f )"1 ( g y ) . S o th e c o m p l e x c a s e a n d th e c a s e o f th e m o d 2 d e g r e e a r e p r o v e d , a n d t h e m u l t i p l i c a t i o n l a w o f o r i e n t a t i o n s s e t t l e s th e o r i e n t e d c a s e .

R e m a r k : W h e n f a n d g h a v e p o s i t i v e i n d e x , t h e m u l t i p l i c a t i o n o f d e g r e e s i s t h e c o b o r d i s m r i n g m u l t i p l i c a t i o n o f R e f . [ 3 9 ] . O n e n e e d s t o s h o w t h a t i f a c o m p a c t c o n n e c t e d s u b m a n i f o l d M o f Y c o n s i s t s o f r e g u l a r v a l u e s f o r a p r o p e r С к+1 - Ф к- т а р f : X -» Y , t h e n f _:L(M ) i s c o b o r d a n t t o f ' 4 y o } x M , f o r a n a r b i t r a r y y o £ M , a n d t h a t i n t h e o r i e n t e d c a s e ( i . e . , M o r i e n t e d s u b m a n i f o l d , X , Y o r i e n t e d m a n i f o l d s , f a C k+1 - Ф * ( Т ) - m a p ) , f ' 1 (M ) a n d f " 1{ y o } x M a r e o r i e n t e d c o b o r d a n t .

N o t e : P r o o f t h a t Ф*(1)Г1 GL(Ë) h a s t w o p a t h - c o m p o n e n t s (E r e a l ) . T h e c a n o n i c a l m a p

L (E . E )q: L(E- E) ■* Е Ж 1)

312 ISNARD

h a s f o r a r e s t r i c t i o n a p r i n c i p a l - G L c ( E ) - b u n d l e G L ( E ) -> q ( ® o ( E , E ) ) ([ 1 0 , 1 1 ,2 8, 4 2 ] ) . S i n c e t h e s e t q ( $ * ( I ) ) i s c o n t r a c t i b l e , i t i s t h e d o m a i n o f a l o c a l s e c t i o n f o r t h a t b u n d l e [ 3 4 , 3 5 ] , w h i c h i s t h e n t r i v i a l o v e r i t [ 2 3 ] . T h i s m e a n s t h a t t h e s e t q"1 ( q ( $ * ( I ) ) ) = Ф*(1) n G L ( E ) i s h o m e o m o r p h i c to q ( ® * ( I ) ) n G L C( E ) , w h i c h h a s t w o p a t h - c o m p o n e n t s ( a c t u a l l y , h a s t h e h o m o t o p i c t y p e o f G L C( E ) ) .

R E F E R E N C E S

[1] ABRAHAM, R ., ROBBIN, J . , Transversality of Maps and Flows» Benjamin, New York (1967).[2] ATIYAH, M. F . , К-Theory, Benjamin, New York (1967).[3] BROWDER, F .E . , "Local and global properties of nonlinear map mappings in Banach spaces", Istituto

N azion aled i Alta M atematica, Symposia M athematica, 2 (1968).[4] BROWDER, F .E . , NUSSBAUM, R .D ., The topological degree for noncompact nonlinear mappings in

Banach spaces, Bull. Am. Math. Soc. 74 (1968) 671.[5] CACCIOPOLI, R ., Sulle corrispondenze funzionali inverse diramate, Rend. Acc. Naz. Lincei 24 (1936)

258; 416.[6 ] CRONIN, J . , Fixed points and topological degree in nonlinear analysis, Math. S u rv ey sN o .il, Am. Math.

So c ., Providence, R.I. (1964).[7] DOUADY, A ., Un espace de Banach dont le groupe linéaire n' est pas connexe, Indag. Math. 68 (1965) 787.[8] DUGUNDJ1, J . , An extension of T ietze ’ s Theorem, Рас. J. Math. 1 (1951) 353.[9] DUNFORD, N ., SCHWARTZ, J . T . , Linear Operators, Interscience, New York (1958).

[10] EELLS, J . , A Setting for Global Analysis, Bull. Am. Math. Soc. 72 (1966) 751.[11] EELLS, J . , "Fredholm structures", Proc. Symp. Pure Math. 18, Am. Math. S o c., Providence, R. 1.(1970).[12] ELWORTHY, K .D . , Fredholm maps and GLC(E) structures. Bull. Am. Math. Soc. 74 (1968) 582.[13] ELWORTHY, K .D ., TROMBA, A ., "Degree theory on Banach m anifolds", Proc. Symp. Pure Math. 18

Am. Math. S o c ., Providence, R.I. (1970).[14] ELWORTHY, K .D . , "Differentiable structures and Fredholm maps on Banach m anifolds", Proc. Symp.

Pure Math. 1 5 , Am. Math. S o c ., Providence, R.I. (1970).[15] GEBA, K . , Fredholm o-proper maps of Banach spaces, Fund. Math. 64 (1969) 341.[16] GEBA, K . , On the homotopy groups of GLC(E), Bull. Acad. Polon. Sci. Math. Astron. Phys. 16 (1968) 699.[17] HUSEMOLLER, D . , Fibre Bundles, Me Graw-Hill, New York (1966).[18] JÀNICH, K. , Vektorraumbiindel und der Raum der Fredholm-Opetatoren, Math. Ann. 161 (1965) 129.[19] KUIPER, N .H ., The homotopy type of the unitary group of Hilbert space, Topology 3 (1965) 19.[20] LANG, S . , Analysis II, Addison-Wesley, Reading, Mass. (1966).[21] LANG, S . , Introduction to Differentiable Manifolds, Interscience, New York (1962).[22] LERAY, J . , SCHAUDER, J. Topologie et équations fonctionelles, Ann. E .N. S. 51 (1934) 45.[23] LIULEVICIUS, A ., Characteristic Classes and Cobordism, Aarhus University, Denmark (1967).[24] NAGUMO, М ., A theory of degree of mappings based on infinitesimal analysis, Am. J . Math. 73 (1951) 485.[25] NAGUMO, M ., Degree of mapping in convex linear topological spaces, Am. J. Math. 73 (1951) 497.[26] NEUBAUER, G . , "Der Homotopietyp der Automorphismengruppe in den Râumen £p and c 0, Math. Ann. 174

(1967) 33.[27] NEUBAUER, G . , "On a class of sequence spaces with contractible linear group", Notes, Berkeley (1967).[28] PALAIS, R .S ., "Homotopy theory of infinite-dimensional m anifolds", Notes, Brandéis University,

Waltham, Mass. (1967).[29] PALAIS, R .S . , Homotopy type of infinite dimensional manifolds, Topology 5 (1966) 1.[30] PALAIS, R .S ., On the homotopy type of certain groups of operators, Topology 3 (1965) 271.[31] SARD, A ., The measure of the critical values of differentiable maps, Bull. Am. Math. Soc. 48 (1942 ) 883.[32] SCHWARTZ, J . T . , Differential Geometry and Topology, Gordon and Breach, New York (1968).[33] SMALE, S . , An infinite dimensional form of Sard’ s theorem, Am. J. Math. _87 (1965) 861.[34] SPAN1ER, E . , Algebraic Topology, Me Giaw-Hill Book Company, New York (1966).[35] STEENROD, N .E . , The Topology of Fibre Bundles, Princeton University press, Princeton, N .J. (1951).[36] STERNBERG, S ., Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, N .J. (1964).[37] SvARC, A. S ., The homotopic topology of Banach spaces, Dokl. Akad. Nauk SSSR 154 (1964) 61.

English Transi. : Am. Math. Soc. Transi. 5 (1964) 57.[38] THOM, R ., "Un lem ma sur les applications différentiables", Bol. Soc. Mat. Мех. (2) 1 (1956) 57.[39] THOM, R ., Quelques propriétés globales des varietés différentiables, Comment. Math. Helv. 28 (1954) 17.

IAEA- SM R-11/36 313

[40] VAINBERG, М. M ., Variational Methods for the Study of Non-Linear Operators", Holden Day, San Francisco, California (1964).

[41] WHITNEY, H ., Geometric Integration Theory, Princeton Univ. Press, Princeton, N .J. (1957).[42] ISNARD, C . , Degree Theory on Banach manifolds, Ph.D. dissertation, The University of Chicago (1972).[43] ISNARD, C . , Introduction to the topological degree (to be published, Atas. Soc. Bras. Mat.)[44] ISNARD, C ., A generalization of the Leray-Schauder degree (to be published).

IAEA-SM R-11/37

EXISTENCE AND NON-EXISTENCE FOR SEMI-LINEAR ELLIPTIC EQUATIONS

J.L. KAZDAN Department of Mathematics,University of Pennsylvania,Philadelphia, P a .,United States of America

Abstract

EXISTENCE AND NON-EXISTENCE FOR SEMI-UNEAR ELLIPTIC EQUATIONS.After a summary of the linear theory, existence and non-existence of a given sem i-linear equation

are studied. Several applications are presented.

1. I N T R O D U C T I O N

S i n c e t h e e x i s t e n c e t h e o r y o f l i n e a r e l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s i s i n a f a i r l y c o m p l e t e f o r m , o n e s h o u l d s e r i o u s l y c o n s i d e r n o n - l i n e a r e q u a t i o n s . T h e q u e s t i o n w e h a v e i n m i n d i s q u i t e p r i m i t i v e : o n t h e a s s u m p t i o n t h a t a l l d a t a a r e s m o o t h — d o e s t h e r e e x i s t , a t l e a s t , o n e s o l u t i o n o f a g i v e n s e m i - l i n e a r e q u a t i o n s u c h a s

Д и + c u = f ( x , u) i n f i , w i t h и = 0 o n 9 f i ( 1 . 1)

w h e r e Я С R n i s , s a y , t h e u n i t b a l l a n d c ( x ) i s s o m e g i v e n s m o o t h f u n c t i o n ? L i t t l e i s k n o w n o n t h i s b a s i c q u e s t i o n .

F o r s i m p l i c i t y , in t h i s p a p e r w e s h a l l l i m i t o u r d i s c u s s i o n to E q . (1 . 1) a n d g i v e s e v e r a l e x a m p l e s . M o s t o f w h a t f o l l o w s e x t e n d s t o L u = f ( x , u) w h e r e L i s a n y s e c o n d - o r d e r u n i f o r m l y e l l i p t i c o p e r a t o r , n o t n e c e s s a r i l y s e l f - a d j o i n t , a n d w i t h v a r i o u s b o u n d a r y c o n d i t i o n s , a s w e l l a s o n c o m p a c t m a n i f o l d s . A m o r e c o m p l e t e d i s c u s s i o n w i t h p r o o f s a n d a d d i t i o n a l r e f e r e n c e s i s t o b e f o u n d i n R e f . [ 6 ] .

Э2 3 2 u nN o t a t i o n : Л и = — r? + . . . + - —я , Î Î C R °-------------- Эх^ Э х *

i s a b o u n d e d d o m a i n w i t h s m o o t h b o u n d a r y , a n d t h e i n n e r p r o d u c t i n L 2( f i )i s d e n o t e d b y < , )> .

2 . S U M M A R Y O F L I N E A R T H E O R Y

T h e l i n e a r e q u a t i o n

Д и + c u = F ( x ) i n f i , w i t h и = 0 o n 9 f i ( 2 . 1)

i s c l a s s i c a l . W e l i s t o n l y t h e f a c t s w e s h a l l r e f e r t o .

315

316 KAZDAN

E x i s t e n c e . I f t h e o n l y s o l u t i o n o f t h e h o m o g e n e o u s e q u a t i o n ( F = 0) i s u = 0 , t h e n g i v e n a n y F t h e r e i s a u n i q u e s o l u t i o n o f E q . ( 2 . 1 ) .

O n t h e o t h e r h a n d , i f t h e r e i s a n o n - t r i v i a l s o l u t i o n o f th e h o m o g e n e o u s e q u a t i o n , t h e n t h i s k e r n e l i s f i n i t e - d i m e n s i o n a l a n d o n e c a n s o l v e E q . ( 2 . 1 ) i f a n d o n l y i f F i s o r t h o g o n a l t o t h e k e r n e l .

T h i s i s , o f c o u r s e , j u s t a s t a t e m e n t o f t h e F r e d h o l m a l t e r n a t i v e .A s i m p l e e x a m p l e i s

u ' 1 + u = F ( x ) o i l Í2 = { 0 < x < 7Г}

w i t h t h e b o u n d a r y c o n d i t i o n s u (0 ) = a(n) = 0 . H e r e s i n x i s a b a s i s f o r t h e k e r n e l o f u' ' + u w i t h t h e g i v e n b o u n d a r y c o n d i t i o n s , s o a s o l u t i o n e x i s t s i f a n d o n l y i f

7Г

J ' F ( x ) s i n x d x = 0

о

A l t h o u g h t h e a b o v e g e n e r a l i z e s t o h i g h e r - o r d e r e q u a t i o n s a n d s y s t e m s , w h a t f o l l o w s h o l d s f o r t h e m o s t p a r t o n l y f o r s e c o n d - o r d e r e q u a t i o n s .

F i r s t e i g e n f u n c t i o n . T h e f i r s t e i g e n v a l u e ( i . e . t h e l o w e s t o n e ) o f

- (Д |p + с <p) = X1p in 0 o n [ I t i s i n s u f f i c i e n t l y w e l l - k n o w n t h a t t h i s g e n e r a l i z e s t o n o n - s e l f - a d j o i n t e l l i p t i c e q u a t i o n s b y t h e K r e i n - R u t m a n t h e o r y o f p o s i t i v e o p e r a t o r s [ 7, s e c t i o n 6 ] a p p l i e d t o t h e c o r r e s p o n d i n g c o m p a c t G r e e n ' s o p e r a t o r . T h e n o n e f i n d s t h a t t h e e i g e n v a l u e w i t h l o w e s t r e a l p a r t i s , i n f a c t , a r e a l n u m b e r a n d t h e c o r r e s p o n d i n g e i g e n f u n c t i o n d o e s n o t c h a n g e s i g n . M o r e o v e r , Хх d e p e n d s c o n t i n u o u s l y o n Í2 a n d i f a d o m a i n u С Q t h e n X ^ io ) > X ^ f i ) . O f c o u r s e , f o r n o n - s e l f - a d j o i n t e q u a t i o n s t h e r e m a i n i n g e i g e n v a l u e s m a y b e c o m p l e x . ]

E x t e n d e d m a x i m u m p r i n c i p l e . A s s u m e t h e c o n s t a n t a < Хг w i t h X x a s a b o v e . I f u s a t i s f i e s

- [ A u + c u ] й o u i n Q , u s O o n

t h e n e i t h e r u = 0 o r e l s e u > 0 i n Q .F o r с s 0 ( s o Х-l > 0) a n d a = 0 t h i s i s t h e c l a s s i c a l s t r o n g m a x i m u m

p r i n c i p l e . A s f a r a s w e k n o w t h e e x t e n s i o n w a s f i r s t u s e d b y D u f f [ 4 ] f o r E q . (2 . 1 ) . I t h a s r e c e n t l y b e e n e x t e n d e d t o n o n - s e l f - a d j o i n t s e c o n d - o r d e r e l l i p t i c o p e r a t o r s b y A m a n n ( s e e R e f s [ 1, 8 ] ) .

3 . P E R T U R B A T I O N T H E O R E M S

T h e l e a s t s u r p r i s i n g r e s u l t s o c c u r w h e n t h e f u n c t i o n c ( x ) i n E q . (1 . 1) i s s u c h t h a t o n e c a n a l w a y s s o l v e E q . ( 2 . 1 ) , i . e . w h e n L = Д + c ( x ) I i s

1AEA-SM R-11/3T 317

" i n v e r t i b l e " . O n e t h e n a n t i c i p a t e s t h a t i f f ( x , u) i s n o t t o o l a r g e t h e n o n e c a n a l w a y s s o l v e E q . (1 . 1 ) . In t h i s d i r e c t i o n t h e f o l l o w i n g t h e o r e m s a r e v a l i d :

I . I f L = A + c ( x ) I i s i n v e r t i b l e a n d | -y | i s s u f f i c i e n t l y s m a l l , t h e n t h e r e e x i s t s a s o l u t i o n o f L u = y f ( x , u, V u ) i n Г2 , u = 0 o n 3!2 ( s e e R e f . [ 3 , p p . 3 7 3 - 7 4 ] ) . T h i s s a m e p r o o f y i e l d s s o l v a b i l i t y f o r -y = 1 i f i n s t e a d o n e a s s u m e s t h a t ÍÍ i s s u f f i c i e n t l y s m a l l . T h e p r o o f i m m e d i a t e l y e x t e n d s t o h i g h e r - o r d e r o p e r a t o r s a n d s y s t e m s .

I I . I f L i s i n v e r t i b l e , f ( x , u , V u ) i s u n i f o r m l y b o u n d e d f o r a l l x e f 2 ,u e C 1 (Я), a n d g ( x , u ,V u ) i s s u f f i c i e n t l y s m a l l t h e n o n e c a n s o l v e

L u + g ( x , u , V u ) u = f ( x , u , V u ) i n Q , u = 0 o n ЗГ2 ( 3 . 1)

H e r e , r o u g h l y s p e a k i n g , o n e w a n t s t o c h o o s e g s o s m a l l t h a t t h e t e r m g u i n E q . (3 . 1) d o e s n o t g e t i n t o t h e e i g e n v a l u e r e g i o n o f L , t h u s m a k i n g L u + g u n o t i n v e r t i b l e . T o b e m o r e s p e c i f i c , i f X N < X N+1 a r e s u c c e s s i v e e i g e n v a l u e s o f L a n d i f t h e r e a r e c o n s t a n t s 7 s u c h t h a t f o r a l l x £ f 2 , u e C 1 (f2)

< TN s g ( x , u , v u ) á T N + 1 < ^ N + 1

t h e n t h e r e i s a s o l u t i o n o f E q . ( 3 . 1 ) . T h i s w a s p r o v e d i n v a r i o u s d e g r e e so f g e n e r a l i t y f o r s e c o n d - o r d e r s e l f - a d j o i n t e q u a t i o n s b y D o l p h , L a n d e s m a n , L a z e r , a n d L e a c h , a n d h a s b e e n e x t e n d e d t o n o n - s e l f - a d j o i n t e q u a t i o n s o f h i g h e r o r d e r a n d t o s y s t e m s b y u s [ 6 ] .

I I I . I f t h e r e i s a c o n s t a n t 7 < X-l s u c h t h a t

l i m s u p - S 7 ( 3 . 2 )|s | - » »

t h e n t h e r e e x i s t s a s o l u t i o n o f E q . ( 1 . 1 ) . T h e m o t i v a t i o n h e r e i s t h e s p e c i a l l i n e a r c a s e f ( x , s ) = a ( x ) s + F ( x ) f o r w h i c h E q . ( 1 . 1 ) r e d u c e s t o t h e l i n e a r e q u a t i o n

Д u + (c - a ) u = F ( x )

w h i c h i s a l w a y s s o l v a b l e i f - a s 7 < X j .E x i s t e n c e u n d e r t h e a s s u m p t i o n ( 3 . 2 ) w a s f i r s t p r o v e d b y S t a m p a c c h i a

[ 9 , s e c t i o n 1 0 ] w h o a l s o a l l o w e d a d d i t i o n a l n o n - l i n e a r i t i e s o n t h e l e f t - h a n d s i d e o f E q . ( 1 . 1 ) , a n d w a s e x t e n d e d to n o n - s e l f - a d j o i n t e q u a t i o n s b y u s [ 6 ] . O u r p r o o f i s s k e t c h e d i n s e c t i o n 5 b e l o w . W e d o n o t k n o w o f a n y g e n e r a l i z a t i o n t o h i g h e r - o r d e r e q u a t i o n s o r s y s t e m s .

A s p e c i a l c a s e i s w h e n с á 0 a n d f ( x , u) = g ( x , u) + h ( x , u ) , w h e r e o n e a s s u m e s t h a t f o r a l l x e f i , s £ R

I g ( x , s ) I S c o n s t a n d - 1ц (x , s ) S 7 < Xa

T h i s c a n e a s i l y b e s e e n t o s a t i s f y E q . ( 3 . 2 ) b y t h e m e a n - v a l u e t h e o r e m . T h e c l a s s i c a l c a s e [ 3 , p p . 3 7 2 - 7 4 ] i s w h e r e 7 = 0 . I n p a r t i c u l a r , t h i s g i v e s e x i s t e n c e i f с á 0 a n d f u ( x , u ) ё 0 .

O n e s h o u l d n o t e t h a t t h e s e t h r e e c o n d i t i o n s f o r e x i s t e n c e o f a s o l u t i o n a l l b e g i n w i t h a l i n e a r o p e r a t o r L t h a t i s i n v e r t i b l e a n d t h e n p l a c e c o n d i t i o n s

3 1 8 KAZDAN

o n t h e n o n - l i n e a r l i t y s o t h a t th e r e s u l t i n g o p e r a t o r d o e s n o t , i n s o m e s e n s e , i n t e r a c t w i t h t h e s p e c t r u m o f L . In c a s e s II a n d I I I , t h i s a l l o w s f o r c e r t a i n l a r g e n o n - l i n e a r i t i e s s i n c e t h e w h o l e s e m i - i n f i n i t e i n t e r v a l X < X j i s n o t i n t h e s p e c t r u m o f L .

I f o n e b e g i n s e i t h e r w i t h a l i n e a r o p e r a t o r L t h a t i s n o t i n v e r t i b l e o r e l s e w i t h l a r g e n o n - l i n e a r i t i e s f ( x , u ) , t h e n e x i s t e n c e i s m o r e s u b t l e a n d e s s e n t i a l l y u n e x p l o r e d . T h e r e a r e s o m e e x i s t e n c e t h e o r e m s a n d s o m e n o n - e x i s t e n c e t h e o r e m s , m o s t o f w h i c h a r e m o t i v a t e d b y t h e l i n e a r t h e o r y , i . e . a r e v a l i d d e s p i t e t h e n o n - l i n e a r i t i e s . T h u s , i n p r a c t i c e , g i v e n a s p e c i f i c e q u a t i o n , o n e h a s t o d e v e l o p n e w d e v i c e s . In t h e r e m a i n i n g s e c t i o n s w e s h a l l c o n t e n t o u r s e l v e s w i t h i l l u s t r a t i n g v a r i o u s p h e n o m e n a u s i n g r a t h e r e l e m e n t a r y p r o c e d u r e s .

4 . M E T H O D S O F P R O V I N G E X I S T E N C E

T h e r e a r e s e v e r a l m e t h o d s o f p r o v i n g t h a t a s o l u t i o n o f a c e r t a i n e l l i p t i c e q u a t i o n d o e s e x i s t :

( i ) c a l c u l u s o f v a r i a t i o n s ,( i i ) s u b - a n d s u p e r - s o l u t i o n s ,( i i i ) m o n o t o n i c i t y ,( iv ) c o n t i n u i t y m e t h o d ,(v) S c h a u d e r f i x e d - p o i n t t h e o r e m ,(v i ) L e r a y - S c h a u d e r d e g r e e .

A l l b u t t h e s e c o n d m e t h o d h a v e b e e n d i s c u s s e d i n r e c e n t t e x t s a n d m o n o g r a p h s , s o w e s h a l l o n l y t r e a t m e t h o d ( i i ) , a n d t h i s o n l y f o r E q . (1 . 1 ) .

L e t u s w r i t e L u = Д и + c u . T h e n t h e e x i s t e n c e t h e o r e m f o r ( i i ) a s s e r t s t h a t i f t h e r e i s a s u b - s o l u t i o n u_, i . e . a s o l u t i o n o f t h e i n e q u a l i t i e s

L u S f ( x , и ) i n f i , и S 0 o n 9 f i ( 4 . 1)

a n d i f t h e r e i s a s u p e r s o l u t i o n u+ s a t i s f y i n g

L u + á f ( x , u +) i n f i , u+ S 0 o n 9 f i ( 4 . 2 )

a n d i f u . S u+ , t h e n t h e r e i s a s o l u t i o n и o f E q . (1 . 1 ) . M o r e o v e r , и s a t i s f i e s u_ s и á u + i n f i .

T h u s , i n t h i s m e t h o d , e x i s t e n c e i s r e d u c e d t o t h e — o c c a s i o n a l l y f o r m i d a b l e — t a s k o f f i n d i n g a - p r i o r i e s t i m a t e s i n t h e f o r m o f s u b - a n d s u p e r - s o l u t i o n s . O n e d e c i d e d v i r t u e o f t h i s m e t h o d i s t h a t i t n e i t h e r a s s u m e s t h a t L i s i n v e r t i b l e n o r d o e s i t r e q u i r e a n y g r o w t h h y p o t h e s i s o n f ( x , u ) . A l t h o u g h o n l y r e c e n t l y e x p l o i t e d , t h i s m e t h o d f o r p r o v i n g e x i s t e n c e i s a c t u a l l y q u i t e o l d [ 3 , p p . 3 7 0 - 7 1 , w h e r e a s p e c i a l — b u t e a s i l y g e n e r a l i z e d — c a s e i s t r e a t e d ] . O n e c a n a l s o e x t e n d t h i s m e t h o d t o c e r t a i n s e c o n d - o r d e r q u a s i - l i n e a r e q u a t i o n s [ 2 ] .

A s a s i m p l e e x a m p l e , c o n s i d e r t h e p r o b l e m o f s o l v i n g

Д и = a ( x ) u - 1 + b ( x ) u e u i n f i , и = 0 o n 9 f i

w h e r e b ( x ) ê c o n s t > 0 a n d a ( x ) i s a r b i t r a r y . E x i s t e n c e i s i m m e d i a t e

IA EA -SM R -11/37 319

u p o n n o t i c i n g t h a t w e c a n l e t u _ (x ) = 0 a n d u +(x ) = l a r g e c o n s t > 0 . N o t e t h a t i f o n e r e w r o t e t h e e q u a t i o n s a s

Д и - a ( x ) u = - 1 + b ( x ) u e u

t h e n i t m i g h t a p p e a r t h a t t r o u b l e , i . e . n o n - e x i s t e n c e , w o u l d o c c u r i f - a ( x ) w a s a n e i g e n v a l u e o f Д , w h e r e a s o u r e x i s t e n c e p r o o f s h o w s t h a t , i n f a c t , t h i s t r o u b l e d o e s n o t o c c u r . T h u s , o n e h a s e x i s t e n c e b e c a u s e o f t h e n o n - l i n e a r i t y .

T h e n e x t t w o s e c t i o n s w i l l g i v e o t h e r a p p l i c a t i o n s o f t h i s m e t h o d .

5 . S K E T C H O F P R O O F O F II I IN S E C T I O N 3

W e f i r s t r e w r i t e t h e h y p o t h e s i s t o r e a d t h a t t h e r e i s a c o n s t a n t s Q > 0 a n d a c o n s t a n t 7 < X x , s u c h t h a t i f | s | ê s fl t h e n f o r a l l x e f 2,

f ( x , s ) ,_ _ _ s 7

N e x t o n e a p p l i e s t h e e x t e n d e d m a x i m u m p r i n c i p l e t o s h o w t h a t s i n c e 7 < Х х , t h e n t h e r e i s a s o l u t i o n v 2 1 i n Q o f

A v + ( c + 7 ) v S 0 i n n , v = c o n s t = m o n ЭГ2

i f m i s s u f f i c i e n t l y l a r g e .W e n o w n o t e t h a t a s u p e r - s o l u t i o n o f E q . (1 . 1) i s u + = s 0v г s 0 , s i n c e

t h e n f ( x , u+) 2 - 7 U+ , s o t h a t

Д u + + c u + á - 7 u + g f ( x , u+)

S i m i l a r l y , o n e l e t s u_ = - u + t o f in d a s u b - s o l u t i o n . T h i s c o m p l e t e s t h e p r o o f t h a t a s o l u t i o n e x i s t s .

6 . A D D I T I O N A L A P P L I C A T I O N S

A n o t h e r a p p l i c a t i o n i s t o s h o w t h a t i f o n e c a n s o l v e

Д и = F ( x ) - g ( x , u) i n Q, и = 0 o n 3f2 (6 . 1)

w h e r e F i s a n y g i v e n f u n c t i o n a n d g ( x , s ) ш 0 ( o r g ( x , s ) s 0) f o r a l l s e R , t h e n o n e c a n s o l v e

A v = F ( x ) - 7 g ( x , v ) i n !2 , v = 0 o n ЭГ2 ( 6 . 2 )

f o r a n y O s 7 S 1 ,T o p r o v e t h i s , s a y g ( x , s ) § 0 ( i f n o t , r e p l a c e и a n d v b y - u a n d - v ,

r e s p e c t i v e l y ) . W e n e e d o n l y f i n d a s u b - a n d a s u p e r - s o l u t i o n o f E q . (6 . 2 ) .A s u p e r - s o l u t i o n i s e v i d e n t l y v + = u . L e t tp b e t h e s o l u t i o n o f Aip = F i n f i , <p = 0 o n 957. T h e n a s u b - s o l u t i o n i s v_ = ip - a , w h e r e t h e c o n s t a n t a i s c h o s e n s o l a r g e t h a t v . s и i n Г2.

320 KAZDAN

O n e c a n u s e t h i s m e t h o d f o r s o m e p r o b l e m s a r i s i n g i n p h y s i c s a n d d i f f e r e n t i a l g e o m e t r y , w h e r e o n e s e e k s a p o s i t i v e s o l u t i o n o f E q . ( 1 . 1) .F o r t h i s s e e [ 6 , s e c t i o n s 4 - 6 ] . A p a r t i c u l a r a n d i n t r i g u i n g p r o b l e m — w h i c h i s s t i l l n o t f u l l y u n d e r s t o o d — i s w h e n o n e c a n f in d a p o s i t i v e s o l u t i o n to

aД и = a u + b u i n f i , и = 0 o n 9 f i

w h e r e a > 1 , a a n d b a r e c o n s t a n t s , a n d f i С R n . I n t h i s c a s e , e x i s t e n c e d e p e n d s o n t h e c o e f f i c i e n t s a , b , o n t h e m a g n i t u d e o f a c o m p a r e d t o t h e d i m e n s i o n n , a n d o n s o m e g e o m e t r i c p r o p e r t i e s o f f i .

A d i f f e r e n t s o r t o f a p p l i c a t i o n i s to a q u e s t i o n p o s e d b y G e l ' f a n d [ 5 , p . 3 5 7 ] . H e a s k e d i f e x i s t e n c e f o r E q . ( 1 . 1) f o r a d o m a i n f i i m p l i e s e x i s t e n c e f o r a l l s m a l l e r d o m a i n s u C f i . U s i n g s u b - a n d s u p e r - s o l u t i o n s , i t i s e a s y t o s e e , f o r e x a m p l e , t h a t i f f ( x , s ) g 0 ( o r i f f ( x , s ) s 0) f o r a l l x e f i , s e R , t h e n t h e a n s w e r i s " y e s " : o n e c a n s o l v e E q . ( l . l ) f o r a n y s m a l l e r d o m a i n u .

F o r t h i s s a m e q u e s t i o n , s i n c e i f и) С f i t h e n X 1 (и) Й A ^ f i ) , t h e c o n d i t i o n o f II I i n S e c t i o n 3 s h o w s a d d i t i o n a l c i r c u m s t a n c e s u n d e r w h i c h s o l v a b i l i t y o f E q . (1 . 1) f o r a c e r t a i n d o m a i n i m p l i e s s o l v a b i l i t y f o r a l l s m a l l e r d o m a i n s .

O n t h e o t h e r h a n d , th e s i m p l e l i n e a r e q u a t i o n

u 1 ' = 1 - и o n f i = {0 < x < 4 r }

w i t h и = 0 o n 9 f i i s a n e x a m p l e w h e r e o n e h a s e x i s t e n c e f o r a d o m a i n f i b u t n o t f o r t h e s m a l l e r d o m a i n u) = { 0 < x < ir) ( s e e S e c t i o n 2 ) . T h u s , i n g e n e r a l , t h e a n s w e r t o G e l ' f a n d ' s q u e s t i o n i s " n o " .

7 . N O N - E X I S T E N C E

T h e s i m p l e s t n o n - e x i s t e n c e r e s u l t s u s e l i n e a r e x i s t e n c e t h e o r y .A s a c r u d e s t a r t , o n e m i g h t c o n s i d e r

Д и + X jU = f ( x , u) i n f i , и = 0 o n 9 f i (7 . 1)

w h e r e X } i s t h e f i r s t e i g e n v a l u e o f Д i n f i . I f a s o l u t i o n и e x i s t s , t h e n , b y t h e l i n e a r t h e o r y , f ( x , u) m u s t b e o r t h o g o n a l t o t h e f i r s t e i g e n f u n c t i o ni-pi ( j u s t t a k e t h e i n n e r p r o d u c t o f E q . ( 7 . 1) w i t h <p a n d i n t e g r a t e b y p a r t s ) . H o w e v e r , s i n c e cp1 d o e s n o t c h a n g e s i g n , t h i s s h o w s t h a t i f f ( x , s ) d o e s n o t c h a n g e s i g n t h e n a s o l u t i o n и d o e s n o t e x i s t — u n l e s s и = 0 .

F o r a m o r e s u b t l e r e s u l t , i t i s u s e f u l t o k e e p t h e e x a m p l e

Д и = t F ( x ) eu i n f i , и = 0 o n 9 f i (7 . 2)

i n m i n d , w h e r e т i s a p a r a m e t e r a n d F i s a g i v e n f u n c t i o n . I f т ê 0 a n d F > 0 , t h e n o n e c a n a l w a y s s o l v e t h i s b y II I o f S e c t i o n 3 ( s i n c e t h e n f ( x , u) = t F ( x ) eu h a s f u ê 0 ) . H o w e v e r , i f т < 0 i s l a r g e a n d n e g a t i v e t h e n o n e h a s n o n - e x i s t e n c e .

T h i s n o n - e x i s t e n c e i s s h o w n a s f o l l o w s . F i r s t c o n s i d e r Д и = f ( x , u)a n d a s s u m e f ( x , s ) á 0 f o r a l l s e R , a n d t h a t f ( x , s ) s f s (x , 0 ) s f o r s a 0 .

IA EA -SM R -11/37 321

L e t X j b e t h e f i r s t e i g e n v a l u e o f

- [ Aq> - f s (x , 0)<p] = Хг <р i n Í2, tp = 0 o n ЭП ( 7 . 3 )

a n d n o t e t h a t w e c a n a s s u m e t h e f i r s t e i g e n f u n c t i o n 0 . I f < 0 w e c l a i m t h e r e i s n o s o l u t i o n o f E q . (1 . 1) .

W e p r o v e t h i s b y c o n t r a d i c t i o n . S a y t h a t t h e r e i s a s o l u t i o n u . T h e n f S 0 a n d t h e m a x i m u m p r i n c i p l e i m p l y u 6 0 . N o w , t a k e t h e i n n e r p r o d u c t o f b o t h s i d e s o f E q . ( 1 . 1 ) w i t h cp t o o b t a i n

- Х ,< (p, u У = ■yip, Д и - f s (x , 0)u >

= ( c p , f ( x , u) - f s (x , 0)u)> S 0

B u t t h i s i s a c o n t r a d i c t i o n s i n c e t h e l e f t - h a n d s i d e i s p o s i t i v e .T o a p p l y t h i s to E q . (7 . 2 ) , o n e n o t e s t h a t i f т i s l a r g e a n d n e g a t i v e

t h e n t h e e i g e n v a l u e X j i n E q . ( 7 . 3 ) i s n e g a t i v e .T h i s c a n r e a d i l y b e e x t e n d e d t o a l a r g e c l a s s o f e q u a t i o n s . W e f o r e g o

t h e e x t e n s i o n , h o w e v e r , a n d t u r n t o a m o r e i n t e r e s t i n g c o n s e q u e n c e o f t h e s e r e s u l t s , n a m e l y , a s s u m i n g F > 0 i n E q . ( 7 . 2 ) , t h e n t h e r e i s a c o n s t a n t - oo < t 0 < 0 s u c h t h a t o n e c a n s o l v e E q . (7 . 2) f o r a l l t > t 0 b u t o n e c a n n o t s o l v e E q . (7 . 2) f o r a n y т < т 0 . T h e e x i s t e n c e f o r a l l т § 0 a n d t h e n o n - e x i s t e n c e f o r s o m e тг , s u f f i c i e n t l y l a r g e n e g a t i v e , w e r e s h o w n a b o v e . E x i s t e n c e f o r т < 0 w i t h | t | s m a l l i s a c o n s e q u e n c e o f I i n S e c t i o n 3 . O n th e o t h e r h a n d , th e f i r s t a p p l i c a t i o n i n s e c t i o n 6 s h o w s t h a t n o n - s o l v a b i l i t y f o r s o m e < 0 i m p l i e s n o n - s o l v a b i l i t y f o r a l l т < a n d c o m p l e t e s t h e p r o o f .

N e e d l e s s t o s a y , t h i s p h e n o m e n o n a l s o e x t e n d s t o a l a r g e c l a s s o f e q u a t i o n s . I n a d d i t i o n , b y a d i f f e r e n t b u t m o r e s p e c i a l p r o o f , o n e c a n e x t e n d t h e n o n - e x i s t e n c e f o r E q . (7 . 2) f o r т l a r g e a n d n e g a t i v e t o c e r t a i n F o f v a r i a b l e s i g n , n a m e l y , i f t h e s o l u t i o n ф o í

Аф = F i n £2, ф = 0 o n 9Q

i s n e g a t i v e s o m e w h e r e , t h e n t h e r e i s n o s o l u t i o n o f E q . ( 7 . 2) f o r т l a r g e a n d n e g a t i v e .

W e s h a l l c l o s e b y m e n t i o n i n g s o m e p e c u l i a r a s p e c t s c o n c e r n i n g f i n d i n g a s o l u t i o n o f t h e d e c e p t i v e l y i n n o c e n t - l o o k i n g e q u a t i o n

Д и = с - k ( x ) e u ( 7 . 4 )

o n a c o m p a c t R i e m a n n i a n m a n i f o l d M ( w i t h o u t b o u n d a r y ) , s e e R e f . [ 1 0 ] .H e r e с i s a c o n s t a n t . N o t e t h a t t h e l i n e a r m a p Д i n E q . (7 . 4 ) i s n o t i n v e r t i b l e s i n c e к е г ( Д ) = { c o n s t a n t s } . T h e t h e o r y o f t h i s e q u a t i o n i s f a i r l y c o m p l e t e i f с < 0 a n d d i m ( M ) a r b i t r a r y . F o r с = 0 , i f d i m ( M ) = 2 t h e n n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s f o r e x i s t e n c e a r e k n o w n , w h i l e i f d i m ( M ) i 3 t h e s u f f i c i e n c y i s a n o p e n q u e s t i o n — a n d i s p e r h a p s t h e m o s t s i m p l e - l o o k i n g e q u a t i o n t h a t i s y e t u n r e s o l v e d . O n t h e o t h e r h a n d , a l m o s t n o t h i n g i s k n o w n

322 KAZDAN

i f c > 0 . W e m e n t i o n o n l y w h a t w e t h i n k i s t h e m o s t u n u s u a l p h e n o m e n o n : i f M = S 2 ( w i th t h e s t a n d a r d m e t r i c ) a n d i f с = 2 , t h e n a n y s o l u t i o n u m u s t s a t i s f y

^ e u V k • V F d A = 0

w h e r e F i s a n y f i r s t - o r d e r s p h e r i c a l h a r m o n i c , A F + 2 F = 0 . In p a r t i c u l a r , t h i s s h o w s n o n - e x i s t e n c e f o r E q . (7 . 4 ) o n S 2 i f с = 2 a n d к = F + c o n s t .


T h i s w o r k w a s i n p a r t s u p p o r t e d b y N S F G r a n t G P 2 8 9 7 6 X . T h i s p a p e r , i n p a r t i c u l a r , i s b a s e d o n j o i n t w o r k w i t h F . W . W a r n e r .

R E F E R E N C E S

[1 ] AMANN, H .t A uniqueness theorem for nonlinear elliptic boundary value problems, Arch. Rat. Mech. Anal. 44 (1972) 178.

[2 ] CHOQUET-BRUHAT, Y . , LERAY, J ., Sur le problème de Dirichlet, quasilinéaire, d’ordre 2, C.R.Ser. A. 274 (1972) 81.

[3 ] COURANT, R ., HILBERT, D ., Methods of Mathematical Physics, 2, Interscience-Wiley, New York (1962).[4 ] DUFF, G .F .D ., Eigenvalues and maximal domains for a quasi-linear elliptic equation, Math. Ann.

131 (1956) 28.[5 ] GEL’FAND, I . М ., Some problems in the theory of quasilinear equations, Usp. Mat. Nauk 14 (1959).

English translation in A. M.S. Translations (2) 29_(1963) 295.[ 6] KAZDAN, J.L., WARNER, F .W ., Remarks on some nonlinear elliptic equations (to appear).[7 ] KREIN, M .G ., RUTMAN, M. A ., Linear operators which leave a cone in Banach space invariant, Usp.

Mat. Nauk £ (1948) 3. English translation in A. M. S. Translation (1) 26 (1950).[ 8] SERRIN, J., A remark on the preceding paper of Amann, Arch. Rat. Mech. Anal. 44(1972) 182.[9 ] STAMPACCH1A, G ., On some regular multiple integral problems in the calculus of variations,

Commun. Pure Appl. Math. 16^(1963) 383.[10] KAZDAN, J. L ., WARNER, F. W ., Curvature functions for compact 2-manifolds, Ann. Math. 99 (1974) 14.

IAEA-SM R-11/38

AN EXAMPLE OF A STRANGE THREE-DIMENSIONAL SURFACE IN (E2

J .J . KOHNIstituto di M atem atica "U. D ini” ,V iale Morgagni,Florence, Italy

Abstract

AN EXAMPLE OF A STRANGE THREE-DIMENSIONAL SURFACE IN C2.After discussion of some well-known geometric properties of pseudo-convex surfaces in (En, an example

of a strange three-dimensional surface in Œ2 is described.

In t h i s p a p e r , w e d e s c r i b e a n e x a m p l e w h i c h w a s f o u n d b y N i r e n b e r g a n d th e a u t h o r ( s e e R e f . [ 3 ] ) . F i r s t w e s h a l l d i s c u s s s o m e w e l l - k n o w n g e o m e t r i c p r o p e r t i e s o f p s e u d o - c o n v e x s u r f a c e s in <Cn .

T o s t a r t w i t h , l e t M b e a d o m a i n in IRm, w i t h a s m o o t h b o u n d a r y S,i . e . w e a s s u m e t h a t t h e r e e x i s t s a r e a l - v a l u e d f u n c t i o n r d e f i n e d o n a n e i g h b o u r h o o d U o f S s u c h t h a t r e C ° (U ) , d r f 0 a n d s u c h t h a t r ( P ) = 0 i f a n d o n l y i f P e S , W e s h a l l t h e n c h o o s e r s o t h a t r < 0 i n s i d e o f M a n d r > 0 o u t s i d e o f M . F o r e a c h P e S w e d e n o t e b y T p (S) th e t a n g e n t s p a c e o f S a t P , i n t e r m s o f l o c a l c o - o r d i n a t e s , e v e r y e l e m e n t o f T P(S) m a y b e w r i t t e n

_m

L ’ L a i Aj= i

w h e r e (a.l t . . . , a m ) e IR m a n d

m

I a J r Xj (P ) = oj=i

H e r e r Xj = d r / d x j . T h e n t h e H e s s i a n o f r a t P , d e n o t e d b y H P ( r ) , i s a q u a d r a t i c f o r m o n T P (S) g i v e n b y

H ( r ) ( L ) = £ r XiXj (P ) a . a . (1)

T h e f o l l o w i n g t h e o r e m i s t h e n k n o w n a n d e a s y t o p r o v e :

T h e o r e m A : M i s ( s t r i c t l y ) c o n v e x w h e n e v e r f o r e a c h P £ S th e q u a d r a t i c f o r m H p ( r ) i s p o s i t i v e ( d e f i n i t e ) .

N o t e t h a t , in th e s p e c i a l c a s e m = 2 a n d r ( x , y) = y - f ( x ) , th e a b o v e t h e o r e m r e d u c e s t o t h e e l e m e n t a r y s t a t e m e n t t h a t c o n v e x i t y i s e q u i v a l e n t

323

3 2 4 KOHN

t o f " S 0 . T h e g e n e r a l t h e o r e m c a n b e r e d u c e d t o t h i s s t a t e m e n t b y i n t e r s e c t i n g S w i t h t w o - d i m e n s i o n a l p l a n e s a n d m a k i n g th e a p p r o p r i a t e l i n e a r c h a n g e s o f c o - o r d i n a t e s .

R e c a l l t h a t c o n v e x i t y o f M i s a l s o c h a r a c t e r i z e d b y t h e e x i s t e n c e o f s e p a r a t i n g h y p e r p l a n e s , i . e . M i s c o n v e x i f a n d o n l y i f f o r e a c h P é S t h e r e e x i s t s a l i n e a r f u n c t i o n w h i c h v a n i s h e s a t P b u t d o e s n o t v a n i s h a t a n v p o i n t i n M .

In t h e t h e o r y o f s e v e r a l c o m p l e x v a r i a b l e s t h e r e i s a n o t io n w h i c h i s v e r y b a s i c a n d w h i c h i s i n m a n y w a y s a n a l o g o u s to c o n v e x i t y . I t i s c a l l e d p s e u d o - c o n v e x i t y a n d w e s h a l l n o w d e s c r i b e i t . L e t M с (Cn b e a d o m a i n w i t h a s m o o t h b o u n d a r y S g i v e n b y a r e a l - v a l u e d f u n c t i o n r a s a b o v e . L e t z i , . . . , z n d e n o t e t h e h o l o m o r p h i c c o - o r d i n a t e s i n <Cn a n d w e s e t x^ = R e ( z ^ ) a n d y ¡ = I m ( z j ) . W e a l s o a d o p t t h e u s u a l n o t a t i o n f o r h o l o m o r p h i c a n d a n t i - h o l o m o r p h i c d e r i v a t i v e s , i . e .

U z i = *u y .

u Z j = i ( U X: + nT - T u v .■M

F o r P e S w e d e n o t e b y T 1,0 t h e s p a c e o f c o m p l e x v e c t o r s o f t y p e (1 , 0) t a n g e n t t o S a t P , i f L i s a n y s u c h v e c t o r i t c a n b e w r i t t e n in t h e f o r m :

L - I * i i f : <31

(P ) = 0

T h e c o m p l e x a n a l o g u e o f th e H e s s i a n i s t h e s o c a l l e d L e v i f o r m w h i c h i s a H e r m i t i a n f o r m f o r e a c h P £ S , on t h e s p a c e T p ' 0 (S ) , i f L e T u (S) g i v e n b y e x p r e s s i o n (3) t h e n t h e L e v i f o r m a p p l i e d t o L i s g i v e n b y

■ b zjZj ( P ) a i a j (4)

M i s c a l l e d p s e u d o - c o n v e x i f t h i s f o r m i s p o s i t i v e s e m i - d e f i n i t e a n d s t r o n g l y p s e u d o - c o n v e x i f i t i s p o s i t i v e d e f i n i t e .

I t i s i m p o r t a n t ( a n d e a s y ) to c h e c k t h a t t h e s e n o t i o n s a r e i n d e p e n d e n t o f t h e c h o i c e o f t h e d e f i n i n g f u n c t i o n r ( i . e . i f r ' i s a n o t h e r f u n c t i o n s u c h t h a t r ' = 0 o n S , d r ' / 0 a n d r ' > 0 o u t s i d e o f M a n d r ' < 0 i n M t h e n th e n u m b e r o f p o s i t i v e , n e g a t i v e a n d z e r o e i g e n v a l u e s o f th e L e v i f o r m r e l a t i v e t o r ' a t P i s t h e s a m e a s t h a t o f th e L e v i f o r m r e l a t i v e t o r a t P ) . F u r t h e r m o r e , t h e s e n o t i o n s a r e i n d e p e n d e n t o f th e c h o i c e o f h o l o m o r p h i c c o - o r d i n a t e s .S o t h a t j u s t a s c o n v e x i t y d e p e n d s o n l y o n t h e l i n e a r s t r u c t u r e s o p s e u d o c o n v e x i t y d e p e n d s o n l y o n t h e c o m p l e x s t r u c t u r e .

IAEA- SM R-11/38 325

O b s e r v e t h a t th e L e v i f o r m i s a n n x n m a t r i x a p p l i e d t o a n ( n - 1 ) - d i m e n s i o n a l s u b s p a c e . C o n s i d e r th e f u n c t i o n R d e f i n e d b y

R = e Xr - 1 (5)

T h i s f u n c t i o n a l s o d e f i n e s th e b o u n d a r y (i . e . d R / 0 , R > 0 o u t s i d e o f M a n d R < 0 i n M ) . F u r t h e r m o r e , i f th e L e v i f o r m i s p o s i t i v e d e f i n i t e a n d i f X i s s u f f i c i e n t l y l a r g e t h e n th e n x n m a t r i x ( R z .2 . ( P ) ) i s p o s i t i v e d e f i n i t e . T o s e e t h i s n o t e t h a t th e L e v i f o r m b e i n g p o s i t i v e d e f i n i t e i m p l i e s t h a t t h e r e e x i s t s a c o n s t a n t С > 0 s u c h t h a t

^ г 2 . г . (P ) a j â j S С | a |2 (6)

w h e n e v e r

£ r z . (P ) a t = 0 (7)

w h e r e

№ Uil2

N o w w e c a n w r i t e (Cm a s a d i r e c t s u m o f t h e s u b s p a c e o f v e c t o r s s a t i s f y i n g c o n d i t i o n (7) a n d t h e o n e - d i m e n s i o n a l s u b s p a c e g e n e r a t e d b y t h e v e c t o r ( r Z l ( P ) , r z 2( P ) , . . . , г г п ( Р ) ) . W e w i s h t o s h o w t h a t t h e r e e x i s t s a c o n s t a n t s u c h t h a t

^ R Zizj (p ) (a i + (s j + b r Zj (p )) - c o n s t ^ I a t + b r z . ( P ) |2 (8)

F i r s t n o t e t h a t

^ | a i + b r Z i ( P ) I2 = ^ l a j 2 + | b I2 ^ | r z . ( P ) |2 (9)

T h e c r o s s t e r m s v a n i s h b e c a u s e o f c o n d i t i o n ( 7 ) . N o w

R z j z j = ( * r z . z . + X2 r z . rg. ) e Xr

s o t h a t t h e r i g h t - h a n d s i d e o f (8) e q u a l s

j j ' - ^ r z i z j ( P ) a i â j + X ^ r ZiZj ( P ) ( a ¡ b r Zj (P ) + b r 2 . (P ) â j )

+ x2 |b|2 X k Zi(P)l2}( 1 0 )

326 KOHN

N o w t h e c r o s s t h e o r e m c a n b e b o u n d e d b y

X ( s m a l l c o n s t | a |2 + l a r g e c o n s t | b | 2 )

C h o o s i n g t h e s m a l l c o n s t a n t s u f f i c i e n t l y s m a l l w e c a n a b s o r b t h e f i r s t t e r m i n th e f i r s t t e r m o f ( 10 ) ( b y v i r t u e o f ( 6)) a n d t h e s e c o n d t e r m c a n t h e n b e a b s o r b e d in t h e t h i r d t e r m o f (10) b y c h o o s i n g X s u f f i c i e n t l y l a r g e . T h u s , u s i n g (9 ) , w e o b t a i n t h e d e s i r e d r e s u l t .

T h e n o t i o n o f s t r o n g p s e u d o - c o n v e x i t y i s , i n m a n y w a y s , a n a l o g o u s to t h e n o t i o n o f s t r i c t c o n v e x i t y . In p a r t i c u l a r , t h e r e a l p a r t s o f h o l o m o r p h i c f u n c t i o n s a r e t h e a n a l o g u e s o f l i n e a r f u n c t i o n s . T h i s i s i l l u s t r a t e d i n th e f o l l o w i n g , w e l l - k n o w n e l e m e n t a r y t h e o r e m :

T h e o r e m B . I f M i s s t r o n g l y p s e u d o - c o n v e x a n d i f P e S , t h e n t h e r e e x i s t s a n e i g h b o u r h o o d U o f P a n d a h o l o m o r p h i c f u n c t i o n h o n U s u c h t h a t h ( P ) = 0 a n d P i s t h e o n l y z e r o o f R e ( h ) w h i c h l i e s in M n u.

P r o o f . B y t h e a b o v e r e m a r k w e c a n c h o o s e a n r s o t h a t t h e m a t r i x ( r z . z . (P ) ) i s p o s i t i v e d e f i n i t e . T h e n w e h a v e , b y T a y l o r ' s t h e o r e m , 1 1

r = R e ( h ) + ^ r z . Z j (P ) Z jZ j + 0 ( | z | 3 ) (11)

h = 2 ^ r 2. (P ) z i + X r z¡2 j (P ) z iZj

i ij

| z I = ( I z ! I2 + . . . + | z u |2 ) *

S i n c e t h e s e c o n d t e r m o n t h e r i g h t h a n d s i d e o f E q . (11 ) i s g r e a t e r t h a n o r e q u a l t o c o n s t | z |2 w e s e e t h a t w h e n e v e r h (z ) = 0 a n d z i s s m a l l a n d f 0 t h e n r i s p o s i t i v e a n d h e n c e z i s o u t s i d e o f M .

C o r o l l a r y . U n d e r t h e s a m e h y p o t h e s i s a s a b o v e t h e r e e x i s t s a n e i g h b o u r h o o d U a n d a h o l o m o r p h i c c o - o r d i n a t e s y s t e m v x , . . . , v n o n U s u c h S n U i s s t r i c t l y c o n v e x r e l a t i v e t o t h e l i n e a r s t r u c t u r e i n d u c e d b y t h e r e a l a n d i m a g i n a r y p a r t s o f t h e v j .

P r o o f : S i n c e (dh) ^ O w e c a n s e t v n = h a n d c h o o s e a h o l o m o r p h i c c o - o r d i n a t e s y s t e m v j , . . . , v n w i t h o r i g i n a t P . T h e n , f r o m e x p r e s s i o n (1 1 ) , w e s e e t h a t t h e T a y l o r e x p a n s i o n o f r , i n t e r m s o f t h e V j , i s

r = R e ( v n) + ^ r v .v . (P ) ViVj + 0 ( I v I3 ) (12)

H e n c e r v ¡v j ( P ) = rv ¡v j (P ) = 0 a n d s i n c e r v .7 . ( P ) i s p o s i t i v e d e f i n i t e w e s e e t h a t t h e r e a l H e s s i a n i s p o s i t i v e d e f i n i t e , a n d t h e p r o o f i s c o n c l u d e d b y t h e o r e m A .

w h e r e

a n d

IA EA-SM R-11/3 8 327

T h e f o l l o w i n g i s a l s o a w e l l - k n o w n r e s u l t i n t h i s f i e l d . H o w e v e r , t h e p r o o f i s s o m e w h a t i n v o l v e d s o w e d o n o t i n c l u d e i t h e r e .

T h e o r e m C . I f P e S a n d t h e L e v i - f o r m i s i d e n t i c a l l y z e r o in S n U , w h e r e U i s a n e i g h b o u r h o o d o f P , t h e n t h e r e e x i s t s a h o l o m o r p h i c f u n c t i o n h d e f i n e d in a n e i g h b o u r h o o d V o f P s u c h t h a t t h e s e t o f z e r o s o f R e ( h ) i s p r e c i s e l y S n v ,

T h e o r e m s В a n d С a n d m a n y o t h e r a n a l o g i e s b e t w e e n c o n v e x i t y a n d p s e u d o - c o n v e x i t y m a k e i t a p p e a r p l a u s i b l e t h a t i f M i s p s e u d o - c o n v e x a n d p e S t h e n t h e r e e x i s t s a h o l o m o r p h i c f u n c t i o n h d e f i n e d i n a n e i g h b o u r h o o d U o f P s u c h t h a t h ( P ) = 0 a n d s u c h t h a t h d o e s n o t v a n i s h i n U П M . I t a l s o s e e m s p l a u s i b l e t h a t t h e r e s h o u l d e x i s t a h o l o m o r p h i c c o - o r d i n a t e s y s t e m on a U s u c h t h a t S n U i s c o n v e x w i t h r e s p e c t to th e l i n e a r s t r u c t u r e i n d u c e d b y t h i s c o - o r d i n a t e s y s t e m .

T h e e x a m p l e b y N i r e n b e r g a n d t h e a u t h o r s h o w s t h a t t h e s e " p l a u s i b l e " s t a t e m e n t s a r e , i n f a c t , n o t t r u e . I n (D2 c o n s i d e r t h e s u r f a c e S g i v e n b y r = 0 w h e r e r i s d e f i n e d b y

r = R e ( w ) + I z 112 + 3 [z 12 R e ( z 10) (13)

F i r s t f o r P £ S w e w i s h t o d e s c r i b e T p , 0( S ) . L e t

T h e n L t a n g e n t to S m e a n s

a r z + b r w = 0

S i n c e f r o m (13 ) w e h a v e r w = \ w e c o n c l u d e t h a t L e T p ' ° ( S ) i f a n d o n l y i f i t i s o f t h e f o r m :

L = a ¿ - 2a r z (P ) ¿ (15)

W e a l s o s e e t h a t r ww = r zw = r wz = 0 a n d r * z = 36 | z | 10 + 3 3 R e ( z 10) . T h e L e v i f o r m o n L i s t h e n (3 6 | z |10 + 3 3 R e ( z 10)) | a |2 w h i c h i s p o s i t i v e e x c e p t a t z = 0 . T h e n w e h a v e t h e f o l l o w i n g r e s u l t :

T h e o r e m D . L e t S b e t h e s u r f a c e in (C2 d e f i n e d b y r = 0 w i t h r g i v e n b y (1 3 ) . L e t U b e a n e i g h b o u r h o o d o f (0 , 0) a n d l e t h b e a h o l o m o r p h i c f u n c t i o n d e f i n e d o n U s u c h t h a t h ( 0 , 0) = 0 . T h e n t h e r e e x i s t t w o p o i n t s ( z i , W i) a n d ( z 2 , w 2 ) in U s u c h t h a t h ( z ¡ , w ¡) = 0 , i = 1, 2 a n d r ( z 1( w i ) > 0 a n d r ( z 2 , w 2 ) < 0.

T h e p r o o f o f t h i s t h e o r e m i s g i v e n i n R e f . [ 3 ] . T o c o n c l u d e w e r e m a r k t h a t t h e a b o v e e x a m p l e w a s d i s c o v e r e d i n c o n n e c t i o n w i t h t h e s t u d y o f l o c a l r e g u l a r i t y p r o p e r t i e s o f t h e i n h o m o g e n e o u s C a u c h y - R i e m a n n e q u a t i o n s , s e e R e f . [ 2 ] .

R E F E R E N C E S

[1 ] GUNNING, R. C ., ROSSI, H . , Analytic Functions of Several Complex Variables, Prentice Hall,Englewood Cliffs, N .J . (1965).

[2] KOHN, J . J . , "Local regularity of ^ on (weakly) pseudo-convex manifolds", J. Diff. Geom. 6 (1972) 523.[3 ] KOHN, J . J . , NIRENBERG, L ., A pseudo-convex domain not admitting a holomorphic support function,

Math. Ann. 201 (1973) 265.

IA EA-SM R-11/39

A CONTINUOUS CHANGE OF TOPOLOGICAL TYPE OF RIEMANNIAN MANIFOLDS AND ITS CONNECTION WITH THE EVOLUTION OF HARMONIC FORMS AND SPIN STRUCTURES

J. KOMOROWSKID epartm ent o f M athem atical Methods in Physics,University o f W arsaw,Warsaw, Poland

Abstract

A CONTINUOUS CHANGE OF TOPOLOGICAL TYPE OF RIEMANNIAN MANIFOLDS AND ITS CONNECTION WITH THE EVOLUTION OF HARMONIC FORMS AND SPIN STRUCTURES.

The quotient topology o f the superspace Л = U x Riem(X)/Diff (the union is taken over all three- dimensional smooth manifolds) does not admit continuous paths between classes represented by Riemannian metrics on non-diffeomorphic manifolds. In this paper, a "very slightly" weaker topology in ^ p e rm ittin g this deficiency to be avoided is introduced. It seems, as was suggested in several papers o f J. A. Wheeler, that a change of the topological type o f Riemannian manifolds can be used to describe creation and annihilation o f pairs of opposite charges, in the non-singular model o f electric field ( i .e . a harmonic 1-form e on a Riemannian three-dimensional manifold with a handle, such that the d u a l*e has only a non-zero period around the handle). The related problems are discussed and some examples and open questions are stated. A relation o f the new topology in Л with a multifold nature of the spin-structure superspace is considered.This gives a more rigorous outlook on the very suggestive figure o f superspace drawn in Wheeler* s "Einstein's Vision".

T h i s p a p e r i s a r e s u l t o f t h e a u t h o r ' s e f f o r t s t o g i v e s o m e m a t h e m a t i c a l f o u n d a t i o n t o a r i g o r o u s d e s c r i p t i o n o f t h e w o n d e r f u l a n d i n s p i r i n g n o t io n o f s u p e r s p a c e i n ' E i n s t e i n ' s V i s i o n ' b y J . A . W h e e l e r .

I N T R O D U C T I O N

L e t H = (S2 , s 0 ) X ( S 1, t 0) b e t h e C a r t e s i a n p r o d u c t o f S 2 a n d S 1 w i t h d i s t i n g u i s h e d p o i n t s s 0e S 2 a n d í q G S 1; i t w i l l b e c a l l e d a h a n d l e . T h e s e t С = ( { s 0 } X S 1) и ( S 2 X { t 0}) i s a s u m o f o n e a n d t w o - d i m e n s i o n a l c y c l e s in t h e h a n d l e H .

L e t X b e a t h r e e - d i m e n s i o n a l 1 C ° ° - m a n i f o l d ; b y M ( X ) , M ( X ) , ( X ) w ed e n o t e t h e o p e n c o n e o f R i e m a n n i a n m e t r i c s o n X , i t s c l o s u r e i n r ° ( è s T * ( X ) ) a n d a s u p e r s p a c e M ( X ) / D i f f (X ) ( s e e R e f s [ 1 , 3]), r e s p e c t i v e l y .

I n s e c t i o n 1 w e g i v e a c o n s t r u c t i o n , n a m e d a p o i n t a d d i t i o n o f t h e h a n d l e , o f a t h r e e - d i m e n s i o n a l C ° ° - m a n i f o l d X & С — w h e r e v d e n o t e s a s e t o f p a r a m e t e r s c h a r a c t e r i z i n g t h e c o n s t r u c t i o n — s u c h t h a t

X & С i s d i f f e o m o r p h i c w i t h X # H

1 All results can be generalized to a number o f dimensions ф 3; for details, see remark 7.

329

3 3 0 KOMOROWSKI

w h e r e X # H i s t h e c o n n e c t e d s u m o b t a i n e d b y r e m o v i n g tw o o p e n t h r e e - d i m e n s i o n a l c e l l s f r o m X a n d H a n d i d e n t i f i c a t i o n o f t h e i r b o u n d a r i e s .

I n s e c t i o n 2 , w e i n t r o d u c e a h o m o m o r p h i s m

k , v „ *n ® T ¥ ( X ) ) 3 u ------ - y e r ( ® T * ( X & C ) )

s u c h t h a t i f g e M ( X ) , t h e n g G M ( X & C ) . T h e i d e n t i f i c a t i o n o f e v e r y g e M ( X ) w i t h g e M (X & C ) i n d u c e s a ' c o n n e c t e d ' t o p o l o g y in M ( X ) u M (X & C ) a n d s o i n U - < ^ ( X &. C ) . T h i s i d e n t i f i c a t i o n m a y b e c o n s i d e r e d a s w i n d i n g o f M ( X ) o n Э M ( X $ C ) ( F i g . 1 ) .

FIG. 1. Winding of M(X) on dM (X& C).

L e t b e t h e s e t o f a l l c l a s s e s o f i s o m o r p h i c ( i s o m e t r i c ) t h r e e - d i m e n s i o n a l R i e m a n n i a n C “ - m a n i f o l d s X . T h e n a t u r a l t o p o l o g y o f ЛИ i s t h a t o f a q u o t i e n t s p a c e , i . e . i n d u c e d b y t h e c a n o n i c a l p r o j e c t i o n s

M ( X ) -------- -

B u t c a n b e a l s o e n d o w e d w i t h t h e t o p o l o g y i n d u c e d b y t h e p r o j e c t i o n s

M ( X ) U M ( X & C ) -------- - „ОС

w h e r e M ( X ) U M ( X & C ) h a s o u r ' c o n n e c t e d ' t o p o l o g y . In s e c t i o n 3 , w e d e f i n e t h i s n e w t o p o l o g y in a n d w e p r o v e i t s ' c a n o n i c a l ' c h a r a c t e r , i . e . i t s i n d e p e n d e n c e o f a r b i t r a r y e l e m e n t s — p o i n t a d d i t i o n s , e t c . — u s e d in t h e d e f i n i t i o n . G o o d p o i n t s o f t h i s t o p o l o g y a r e e x p o s e d i n t h e o r e m 3 a n d r e m a r k 4 .

W e w a n t t o e m p h a s i z e t h a t s u c h r e s u l t s a r e n o t o b t a i n a b l e i f o n e i s g o i n g t o d e a l w i t h a l l X # H i n s t e a d o f o n l y w i t h a l l X & C .

A c o n n e c t i o n o f o u r t o p o l o g i c a l c o n c e p t s w i t h a d e s c r i p t i o n o f a n n i h i l a t i o n o f c h a r g e s i s c o n s i d e r e d i n s e c t i o n 5 .

IA EA-SM R-11/39 331

L e t S 2 a n d S 1 b e e n d o w e d w i t h t h e i r n a t u r a l m e t r i c s . W e d e f i n e s u b s e t s

U * = { ( s , t ) e H : d ( s , s 0 ) < а } э { s 0 } x S 1

U 2 = l ( s , t ) e H : d ( t , t 0 ) < a } d S 2 X { t Q}

w h e r e 0 < a < тт. F o r e v e r y 0 < a , (3 < it U * U U 2 i s a n e i g h b o u r h o o d o f С i n H a n d 3 U ¿ , 3 Ug a r e t w o - d i m e n s i o n a l C ~ - s u b m a n i f o l d s i n H .

D e f i n i t i o n 1 . L e t £2 b e a n o p e n n e i g h b o u r h o o d o f С i n H s u c h t h a t 3 S"2 i s at w o - d i m e n s i o n a l C “" - s u b m a n i f o l d in H , d i f f e o m o r p h i c w i t h S 2. L e t к b e a d i f f e o m o r p h i s m

Э П Х ] 0 , 1] Э ( s , t ) ----------► K ( s , t ) e ñ \ C

s u c h t h a t : 1 ) к ( . , 1 ) = i d e n , 2 ) t h e r e e x i s t s 0 < a0< it s u c h t h a t f o r e v e r y a,0 < a < a0, a v e c t o r f i e l d F o n £ 2 \ C , d e f i n e d a s

F ( k ( s . t ) ) ( s , t )

i s t r a n s v e r s a l t o 3 Uj, a n d o u t g o i n g f r o m U ¿ , i = 1 , 2 . T h e p a i r (Q, к ) i s c a l l e d a r a d i a l c h a r t o n H . T h e e x i s t e n c e o f r a d i a l c h a r t s i s e a s i l y s e e n ( F i g . 2 ) .

L e t X b e a t h r e e - d i m e n s i o n a l C ° ° - m a n i f o l d a n d l e t b e g i v e n g e M ( X ) .I f Sfp = Ж ( 0 , p ) c T XQ ( X ) i s s u c h t h a t t h e m a p e x p : 5fp -* X i s a n o r m a l c h a r t t h e n Жр i s c a l l e d a d o m a i n o f e x p a t x 0E X ,

L e t u s t a k e a r a d i a l c h a r t ( S 7 ,k ) o n H a n d a d o m a i n , !Xp, o f e x p a t x 0e X ,w h e r e X h a s a f i x e d R i e m a n n i a n m e t r i c g e M ( X ) . N o w , l e t b e g i v e n :1 ) a d i f f e o m o r p h i s m у : 3Q-> 3 p r e s e r v i n g t h e p o s i t i v e o r i e n t a t i o n s ;

3Í2 a n d 3 .5^ h a v e t h e c a n o n i c a l p o s i t i v e o r i e n t a t i o n s d e f i n e d b y t h e R i e m a n n i a n s t r u c t u r e s o f H a n d X , r e s p e c t i v e l y ,

2 ) a C ”° - f u n c t i o n f ( F i g . 3 ) : ] 0 , 1 ] ->■ [ 0 , + oo[ s u c h t h a t f ( ] 0 , i ] ) = 0 a n d f 1 > 0 o nH , 1 ] , f ( 1 ) ê p .

1. THE POINT ADDITION O F THE HANDLE

FIG. 2 . Radial chart.

332 KOMOROWSKI

FIG.3. Function f.

I f pf = s u p f _1(] 0 , p [ ) , t h e n f (] 0 , pf ] ) С [ 0 , p] . L e t Q f = к (9 f i X ] 0 , p f [ ) U C . f i f i s a n e i g h b o u r h o o d o f С i n H . W e d e f i n e a m a p v : f i f -* X a s

v (к ( s , t ) ) = e x p f ( t ) y ( s ) , ( s , t ) £ 3 i î X ] 0 , pf [

v (p ) = x , p e C

O b v i o u s l y , v i s a C " - m a p , I t w i l l b e c a l l e d a p o i n t a d d i t i o n o f С t o X a tx 0 e X . T h e n e x t d e f i n i t i o n e x p l a i n s t h i s t e r m i n o l o g y . T h e s e t{ ( f i , к ) , g, SCp, У, f } w i l l b e c a l l e d p a r a m e t e r s o f t h e p o i n t a d d i t i o n 2 v .

R e m a r k 1 . v _1 ( ( e x p 5 i ^ ) \ { x 0}) = к ( 9 f i X ] pf [ ) i s t h e b i g g e s t s u b s e t o f f i s u c h t h a t v r e s t r i c t e d t o t h i s s u b s e t — l e t t h i s r e s t r i c t i o n b e d e n o t e d b y — i s a d i f f e o m o r p h i s m o n t o i t s i m a g e .

D e f i n i t i o n 2 . L e t v b e a p o i n t a d d i t i o n o f С t o X a t x oe X , w h e r e X i s a R i e m a n n i a n m a n i f o l d w i t h m e t r i c g e M ( X ) . T h e p a r a m e t e r s o f v a r e { ( f i , к ) , Жр, y, f } . W e d e f i n e a C ^ - m a n i f o l d X & С a s f o l l o w s :1 ) X & С i s a t o p o l o g i c a l s p a c e e q u a l t o ( ( X \ { x 0}) w h e r e J ? i s t h es m a l l e s t e q u i v a l e n c e r e l a t i o n s u c h t h a t

( x ~ p) •4------ »■ ( x = v (p ) )

f o r x e X \ { x Q}, p e f i j . ;2 ) C ° ° - m a n i f o l d s t r u c t u r e o n X & C i s g i v e n b y t h e c a n o n i c a l m a p s

i n : f i f ---------► X & C

i x : X \ x 0 ---------- X & C

It i s n e c e s s a r y to s h o w t h a t s o d e f i n e d X & С i s r e a l l y o f c l a s s C “ . S i n c e ( I m i j j ) n ( I m i x ) = 1п ( у _1( ( е х р { x 0})) i t i s s u f f i c i e n t to p r o v e t h a t t h e m a p i x o i n r e s t r i c t e d t o v~l ( ( e x p J Y p ) \ { x 0}) i s a d i f f e o m o r p h i s m o n t o i t s i m a g e . B u t t h i s r e s t r i c t e d m a p i s e q u a l t o v0. T h u s t h e r e s t i s c o n t a i n e d in r e m a r k 1 .

2 In the following ,if we speak about a point addition of С to a Riemannian manifold X with a metric g CM (X), we shall write { (Q , к), y, f } instead o f {(S3, к), g, y , f} .

IA EA -SM R -11/39 333

4G .4 , Construction o f manifold X&C.v

T h e a b o v e c o n s t r u c t i o n o f t h e m a n i f o l d X & С l i e s in t h e r e p l a c e m e n t o f x 0G X b y С w i t h i t s n e i g h b o u r h o o d f tf . T h i s m a y b e i l l u s t r a t e d b y F i g . 4 , w h e r e i s d o t t e d a n d к (dQ X ] | , pf[ ) i s h a t c h e d . T h e h a t c h e d s u b s e t s o f X a n d a r e i d e n t i f i e d .

It i s s e e n t h a t X & С i s d i f f e o m o r p h i c w i t h X # H ; a p o i n t a d d i t i o n i s a k i n d o f t h e c o n n e c t e d s u m .

2 . A T R A N S P O R T O F C O V A R I A N T T E N S O R F I E L D S F R O M X O N T O X

L e t у b e a p o i n t a d d i t i o n o f С t o X a t x Qe X . T h e c a n o n i c a l m a p ж X & С -»• X d e f i n e d a s

V

i x (У) - y e l m i x

^ „ ( y ) =■

^ o i ^ ( y ) . y e t o i a

i s o f c l a s s C ” . T h u s w e h a v e a m o n o m o r p h i s m

7Г*: Г “ ( ® T * ( X ) ) ---------► Г “ ( ® T * ( X & С ) ) (1)

W e s h a l l d e n o t e u = тг*и, и £ Г го( ® T * ( X ) ) .

VV - 1 и

R e m a r k 2 . I t i s s e e n t h a t ы = 0 o n if j (v ( x Q) ) C X & C . T h u s s u p p шC l m i x . In o t h e r w o r d s ,

w (У) =r ( ix1) * u ( i x (У ) ) . y £ I m i x

. yÎm i*

3 3 4 KOMOROWSKI

S o , e v e r y p o i n t a d d i t i o n o f С to X a t x 0e X p o i n t s o u t a w a y o f t r a n s p o r t o f c o v a r i a n t t e n s o r f i e l d s f r o m X o n t o s o m e h o m o l o g i c a l l y d i f f e r e n t m a n i f o l d . N o w w e a r e g o i n g t o d i s c u s s a n i n t e r d e p e n d e n c e o f t w o s u c h t r a n s p o r t s g i v e n b y t w o p o i n t a d d i t o n s o f С t o X a t x Qe X .

L e t v a n d v b e p o i n t a d d i t i o n s o f С t o X a t x 0e X . _It f o l l o w s f r o m R e m a r k 2 t h a t t h e s u p p o r t o f ui £ Г ” ( 0 T * ( X & C ) ) ( о г £ е Г " ( 0 T * ( X & C ) ) r e s p e c t i v e l y ) i s c o n t a i n e d in I m i x ( o r I m f ^ ) ; h e r e Г х : X \ { x 0}-^ X ^ , C i s t h e a n a l o g u e o f i x : X \ { x 0}-> X ^ C .

I t i s o b v i o u s t h a t t h e d i f f e o m o r p h i s m

фо = i x ° V 1: I m i x -------- *■ I m i x

i s s u c h t h a t, * ~ _ v T $ ¿ u = u o n I m i x

I f t h e d i f f e o m o r p h i s m Ф 0 h a d a n e x t e n s i o n to a d i f f e o m o r p h i s m f r o m X & С o n t o X ^ C , t h e n t h e s t r u c t u r e s ( X ^ C , u ) a n d ( X 4 . C , &) w o u l d b e c o n s i d e r e d i s o m o r p h i c 3 . H o w e v e r , t h e r e e x i s t s — in g e n e r a f — b u t a h o m e o m o r p h i s m f r o m X fy С o n t o X & С w h i c h e x t e n d s Ф 0.

N e v e r t h e l e s s , w e h a v e t h e f o l l o w i n g

F u n d a m e n t a l t h e o r e m . L e t X b e a 3 - d i m e n s i o n a l R i e m a n n i a n C “ - m a n i f o l d w i t h a m e t r i c g e M ( X ) . I f v, ¡7 a r e p o i n t a d d i t i o n s o f С t o X a t x 0£ X , у £ Г “ ( ® T * ( X ) ) a n d k e N , t h e n t h e r e e x i s t s a s e q u e n c e o f d i f f e o m o r p h i s m s

----- “ X& c

-------- ► 0n -»■

i n t h e t o p o l o g y o f Г к( ® T * ( X & C ) ) .T h e p r o o f w i l l b e g i v e n l a t e r .

Фп: X & Cs u c h t h a t

Ф'1' OJ - (0

3 . A N E W T O P O L O G Y

B y .Ж w e d e n o t e t h e s e t o f a l l c l a s s e s o f i s o m o r p h i c ( i s o m e t r i c ) t h r e e - d i m e n s i o n a l R i e m a n n i a n C “ - m a n i f o l d s . O r i g i n a l l y , t h e t e r m s u p e r s p a c e r e f e r r e d t o a s in R e f . [ 3 ] , n o w i t m e a n s ^ ( X ) a s i n R e f . [ 1 ] . In o t h e r w o r d s , l e t E b e t h e s e t o f a l l t h r e e - d i m e n s i o n a l C “ - m a n i f o l d s ; t h e n

U

w e a l s o d e f i n e

и M ( X )

J = x e £ / „' D i f f

3 V VIn applications, u> can be a Riemannian metric on X; then (X& C , w) and (X& C , oj) are ’ metric1

v v vstructures _ the metrics are given by the degenerated tensor fields w and и (see remark 2).

IA EA-SM R-11/39 335

W h e n w e w r i t e r o r _ / f w e m e a n t h a t t h e s e s p a c e s a r e e n d o w e d w i t h t h e q u o t i e n t t o p o l o g i e s .

F o r e v e r y X e E w e h a v e c a n o n i c a l m a p p i n g s

M ( X ) — Î— ^ f ( X ) — Í—

a n d

M ( X ) — -— ► Z t í ( X ) ----

w h e r e i ' s a r e s u r j e c t i o n s a n d j ' s a r e i n j e c t i o n s . M o r e o v e r , t h e m a p j i s a h o m e o m o r p h i s m o n t o i t s i m a g e b e i n g a c o n n e c t e d c o m p o n e n t o f , & o r ~j%, r e s p e c t i v e l y . I f g e M ( X ) , t h e n w e d e n o t e [ g ] = j o i (g )

R e m a r k 3 . It i s e a s i l y s e e n t h a t j o i i s a n o p e n m a p .L e t u s i n t r o d u c e t h e f o l l o w i n g d e n o t a t i o n

s : = ( X , g , x , v )

w h e r e X e E , g e M ( X ) , x C X , v i s a p o i n t a d d i t i o n o f С to t h e R i e m a n n i a n m a n i f o l d ( X , g) a t t h e p o i n t x e X .

W e d e f i n e a f a m i l y ( E g ] ) o f s u b s e t s i n Л

^ ( [ g ] ) : = { u u ( V a « f ) : U £ ^ ( [ g ] ) , V e v F ( [ g ] ) }

w h e r e ^ ( [ g ] ) ( o r ^ ( I g l ) , r e s p e c t i v e l y ) , i s a b a s e o f n e i g h b o u r h o o d s o f t h e p o i n t [ g ] e ..«f ( o r [ g ] e , r e s p e c t i v e l y ) . W e r e c a l l t h a t g e M ( X & C ) .

W e a r e g o i n g t o d e f i n e a n e w t o p o l o g y in Л t a k i n g ([ g ] ) t o s t a t e a b a s e o f n e i g h b o u r h o o d s o f t h e p o i n t [ g ] e (W e s h a l l d o t h a t f o r e v e r y p o i n t o f ) B u t n o w w e s t a t e f e w p r o p e r t i e s o f s o d e f i n e d f a m i l y &>% ([ g ] ) .

P r o p o s i t i o n 1 . L e t X , Y e E , <p: Y -* X b e a d i f f e o m o r p h i s m , S j = ( X , g , x , v ), w h e r e t h e p a r a m e t e r s o f t h e p o i n t a d d i t i o n v a r e { ( f i , к ) , 5Tp , y, f } . I f w e d e f i n e s 2: = ( Y , <p''g, <p'1( x ) , ц ) , w h e r e t h e p a r a m e t e r s o f м a r e { ( f i , к ) , 5 f p , (<р~г)ьО у , f } , t h e n f o r e v e r y *2/ e ([ g ] ) t h e r e e x i s t s ^ ' e ([ g ] ) s u c h t h a t С •%/. T h e p r o o f w i l l b e g i v e n l a t e r .

P r o p o s i t i o n 2 . L e t S j = ( X , g , x , v ) , s 2 = ( X , g , x , iu), w h e r e v a n d ц a r e p o i n t a d d i t i o n s o f С t o t h e R i e m a n n i a n m a n i f o l d ( X , g ) a t x e X , t h e n f o r e v e r y ^ e ^ s 2 ([ g ] ) t h e r e e x i s t s W 'e . ([ g ] ) s u c h t h a t W 'C . ‘i/ .T h e p r o o f , b a s e d o n t h e f u n d a m e n t a l t h e o r e m , w i l l b e g i v e n l a t e r .

T h e a b o v e t w o p r o p o s i t i o n s c a n b e c o m b i n e d i n t o

T h e o r e m 1 . L e t X , Y e E a n d <p : Y -» X b e a d i f f e o m o r p h i s m .I f S j = ( X , g , x , v ) , s 2 = ( Y , ip* g, <pml(x ) , ц ) , w h e r e v ( o r ¡л, r e s p e c t i v e l y )

i s a p o i n t a d d i t i o n o f С t o t h e R i e m a n n i a n m a n i f o l d ( X , g ) ( r e s p . ( Y , g ) )a t t h e p o i n t x e X ( o r сp ' 1 (x ) £ Y , r e s p e c t i v e l y ) , t h e n f o r e v e r y < ? / e ([ g ] )t h e r e e x i s t s ([ g ] ) s u c h t h a t W С W. 2

3 3 6 KOMOROWSKI

N o w w e c a n d e f i n e a n e w t o p o l o g y i n L e t u s t a k e1 ) 9C = { X J a g A s u c h t h a t

( X ^ i s d i f f e o m o r p h i c t o X g ) « = * . (a = I3)

( X £ E ) = > ^ X i s d i f f e o m o r p h i c to X „ ^ " ( 2 )

2 ) IN = b a g X} o 6 A " а s e t o f p o i n t a d d i t i o n s ; ^ a g x i s a p o in t

xCXc, g£M(Xa)a d d i t i o n o f С t o X a a t x £ X o , w h e r e X a i s e q u i p p e d w i t h m e t r i c g .T h e a b o v e t w o f a m i l i e s , âT a n d IN , d e t e r m i n e t h e f o l l o w i n g s e t :

S = { s = ( X , g , X, V ) : X e a r , g e M ( X ), x e X . i/ v e l N }â 'IN gx ' a ’ ° o g x ' a b ' a ' « ’ o-gx

a l l o w s u s t o d e f i n e a t o p o l o g y , , i n Л a s f o l l o w s :i f m e . . ^ t h e n a s a b a s e o f n e i g h b o u r h o o d s o f t h e p o in t m w e t a k e

* ( m ) = { x e X „ ? / « : ^ x e ^ x ( m )

w h e r e1 ) X 0 i s t h e o n l y e l e m e n t o f 3C s u c h t h a t t h e r e e x i s t s g e M ( X a ) f o r

w h i c h [ g ] = m ,

s gx £ ®áT]N ’I t i s s e e n t h a t i n t h i s d e f i n i t i o n g i s n o t d e t e r m i n e d u n i q u e l y . H o w e v e r , in t h e l i g h t o f t h e o r e m 1 , o u r d e f i n i t i o n o f t h e t o p o l o g y N i s n o t i n f l u e n c e d b y t h i s a m b i g u i t y .

T h e n e x t t h e o r e m t e l l s u s a b o u t a ’ c a n o n i c a l ’ c h a r a c t e r o f t h e i n t r o d u c e d t o p o l o g y i n Л :

T h e o r e m 2 . L e t b e g i v e n t w o p a i r s , w i t h t h e p r o p e r t i e s l i s t e d in r e l a t i o n s (2 ) :

^ = { Х Л £ А » = 1 v U ax e x a g e m (x„)

a n d

^ = < Y a}oGA M = { ^ agy}aeAy e ï ag £ M ( ï „ )

T h e n

arN » m

P r o o f . O n a c c o u n t o f t h e s y m m e t r y o f o u r p r o b l e m i t i s s u f f i c i e n t t o s h o w t h a t f o r e v e r y m e . f f a n d e a c h o f i t s - n e i g h b o u r h o o d s “X t h e r e e x i s t sa n n e i g h b o u r h o o d СУ ' o f m s u c h t h a t <X ’ С <X'. H e n c e i t i s s u f f i c i e n t t o p r o v e t h a t f o r e v e r y s 2 = ( Y a , h , y , f j a l l y ) e S ^ M a n d e v e r y W ( [ h ] ) t h e r ee x i s t a n d e - ^ si ([ h ] ) s u c h t h a t *&/' С <?/. T o f i n d s u c h w eh a v e t o n o t i c e t h a t t h e r e e x i s t X fl e ^ a n d a d i f f e o m o r p h i s m ф : X g -► Y a .I f w e d e f i n e s j = ( X s , ф *h , ф '1 ( y ) , (y) ) e , t h e n t h e e x i s t e n c e o f t h es o u g h t W f o l l o w s f r o m t h e o r e m 1 .

IA EA-SM R-11/39 3 3 7

T h u s w e h a v e p r o v e d t h a t t h e n e w t o p o l o g y ■ s- m i n - ¿ f i s i n d e p e n d e n t o f t h e a u x i l i a r y e l e m e n t s SC a n d IN. I t s r e l a t i o n t o t h e n a t u r a l ( q u o t i e n t ) t o p o l o g y i n i s g i v e n i n t h e f o l l o w i n g o b v i o u s

T h e o r e m 3 . T h e t o p o l o g y i s w e a k e r t h e n t h e n a t u r a l ( q u o t i e n t ) t o p o l o g yi n ..<!(. H o w e v e r , i f X e E , t h e n b o t h t o p o l o g i e s r e s t r i c t e d t o [ M ( X ) ] C „ < f c o i n c i d e .

A n a d v a n t a g e o f t h i s n e w t o p o l o g y i n „ЛС c a n b e e x p r e s s e d a s

R e m a r k 4 . T h e s e t [ M ( S 3)] U [ M ( S 2 X S 1)] С is :1 ) n o t c o n n e c t e d — h a s t w o c o n n e c t e d c o m p o n e n t s — i f i s e n d o w e d w i t ht h e n a t u r a l ( q u o t i e n t ) t o p o l o g y ,( 2 ) c o n n e c t e d i f i s e n d o w e d w i t h t h e t o p o l o g y

M o r e o v e r , l e t X e L a n d X ( p ) = X # ( S 2 X S 1) § Рн.т “ j¡ ( S 2 X S 1) t h e n

U [ М ( Х ( р ) ) ] С „ i f i s c o n n e c t e d i f i s e n d o w e d w i t h t h e t o p o l o g y ■7~аг-ы . p = о

4 . T H E P R O O F S

P r o o f o f t h e f u n d a m e n t a l t h e o r e m . L e t { ( Q , k ) , 5 Г , 7 , f } , { ( í 2 , k ) , 7 , f }b e t h e p a r a m e t e r s o f t h e p o i n t a d d i t i o n s 7 a n d 7 , r e s p e c t i v e l y . W e n o w h a v e a c o m m u t a t i v e d i a g r a m a s s h o w n in F i g . 5 .

A . L e t u s n o t i c e t h a t i f x e l m i x f i l m i ^ , t h e n 1 ) t h e r e e x i s t s ( s , t ) e3 Í 2 X s u c h t h a t x = i f io k ( s , t ) , 2 ) x i s m a p p e d b y Ф0 = i ^ i ^1 o n t o$ o ( x ) = i x o i x (X ) , J i ñ ^ ' ^ i ' o i ñ 1 (x ) = i ñ o v ^ o v (к ( s , t ) ) = ijj y _ 1 ( e x p f ( t ) -y ( s ) ) = % o k ( T ' 1o 7 ( s ) , f ' - ^ f ( t ) ) .

B . L e t r = \ m i n ( p f , p f ) , U ¡ = i x ( X \ e x p ) C X & C , i = 1 , 2 , a n d V = C U 2.I t i s s e e n t h a t U g C U j a n d V c l m i fi ( F i g . 6 ) . W e s h a l l d e f i n e Фп g i v i n g i t sr e s t r i c t i o n s to t h r e e d i s j o i n t s u b s e t s U 2, V \ i n ( C ) , i n ( C ) w h i c h c o v e r X & C .T h u s w e s t a t e Ф П| Т1 = Фо | TI , ®n|. = i f í o i ñ 1 I. 1ГЛ ■ M o s t o f t h e p r o o f i su2 'U2 '>íí(c ) ‘ n '0 )d e v o t e d t o a c o n s t r u c t i o n o f Ф п | „ . . , „ Ч= ’ФП-

C . T h e d o m a i n o f Ф п i s V \ i fi ( C ) = i ^ o к ( 9 П X ] 0 , f _1 ( 2 r ) [ ) . S i n c e Ф „ h a s t o b e a b i j e c t i o n ( o n t o X & C ) i t i s n e c e s s a r y t h a t I m -i>n = ¿ ¡ ¡ o k ( 8 Î Ï X ] 0 , f " " 1 ) ! ) .

T h u s w e m a y d e f i n e ijj-i о ф по 1а « фп i n s t e a d o f ^ . T h e n e x t f e w p a r t s o f t h e p r o o f c o n t a i n s a c o n s t r u c t i o n o f : ^ n .

FIG. 5. C om m utative d iagram .

3 3 8 KOMOROWSKI

FIG. 6. Illustration of item B.

D . W e p u t Ф п = Фо o n U i \ U 2 . S o w e h a v e n o t r o u b l e s w i t h a s m o o t h s t i c k i n g o f Ф п a n d Ф п | и 2 . L e t u s n o t i c e t h a t U i \ Ü 2 = i n o к (9 Í2 X [ f _1( r ) , f " 1( 2 r ) [ ) . H e n c e , f o r ( s , t ) e 9Г2 X [ f _1( r ) , f _1( 2 r ) [ w e h a v e

^П° ‘ ! s . t ) = « ( ï ' 1 0 T ( s ) l f _1of ( t )) (3)E . T h e m a p к ( o r к , r e s p e c t i v e l y ) , g e n e r a t e s o n f ¿ \ С ( o r f i \ C , r e s p e c t i v e l y ) a v e c t o r f i e l d F ( o r F , r e s p e c t i v e l y ) , t h e f l o w o f w h i c h i s x T ( / c ( s , t ) ) =к ( s , t + t ) ( o r x T (k (s , t ) ) = k ' î s , t + t ) , r e s p e c t i v e l y ) . T h u s , w e h a v e к ( s , t ) = X ^ j Í K f s , 1 ) ) = X j . j i s ) a n d t h e a n a l o g i c a l e x p r e s s i o n f o r к'. N o w (3 ) h a s th e f o r m

Фп ° K ( s , t ) = Х~_г ( y -1 о 7 ( s ) )f о f(t) - 1

F . W e s h a l l m a k e u s e o f t h e f o l l o w i n g l e m m a :L e t E b e t h e s e t o f a l l s u c h f u n c t i o n s f e C “ (IR1) t h a t1 ) f = 0 o n ] - oo, 0 ]2 ) f 1 > 0 o n ] 0, +oo[( H e n c e , f o r e v e r y j e N , t h e r e e x i s t s a¡ > 0 s u c h t h a t f ® i s n o n - d e c r e a s i n g o n [ 0, o-j ] ).

L e m m a 1 . I f f , f e E , 0 < a e I R 1, k e N , t h e n t h e r e e x i s t s a s e q u e n c e F n e S s u c h t h a t1 ) f o r e v e r y n £ N F n = f o n [ a , + oo[,2 ) f o r e v e r y n e N t h e r e e x i s t s t > 0 s u c h t h a t F n = f o n ] - oo, t ] ,3 ) DF n - f [I c k + l (Ri j п- Т Т 1 Г 0 .

It f o l l o w s f r o m t h i s l e m m a t h a t t h e r e e x i s t s a s e q u e n c e F n e C “ (] 0 , 1 ] ) s u c h t h a t1 ) f o r e v e r y n e N F n = f o n [ f ‘ 1( r ) , 1 ] , ^2 ) f o r e v e r y n e N t h e r e e x i s t s t > ^ s u c h t h a t F n = f o n ] 0 , t ] ,

3) II Fn - f II Ck + 1( ]0 , ll) n Z, + „ * 0

w h e r e f , f a r e p a r a m e t e r s o f t h e p o i n t a d d i t i o n s v a n d v, r e s p e c t i v e l y .W e d e f i n e a s e q u e n c e u ne C°°( ] 0 , 1 ] )

IA E A -S M R -ll/3 9 339

Х ( Э П , Е 0) aft an

А,

с

FIG. 7. Illustration o f item G .

A s t h e n e x t s t e p i n t h e c o n s t r u c t i o n o f ip w e pu t

f o r ( s , t ) e 9 Q X f " 1 ( 2 r ) [ , w h a t i s c o m p a t i b l e w i t h r e l a t i o n ( 3 ) .

G . S i n c e к , к a r e r a d i a l c h a r t s o n H t h e r e e x i s t t w o o p e n n e i g h b o u r h o o d s , A i a n d А г , o f С С Г 2 П £ 2 s u c h t h a t1) A2 C A 2, ЭАх^ЭА2 = Ф,2) a A j C K (3Í2 X ] 0 , | [ í n í c ^ ñ X ] 0 , i [ ) , _3) f o r e v e r y p e A x a n d a, /3 ê 0 ( a + j3 0) = ► ( t t F ( p ) + | 3 F ( p ) ^ 0 ) ,w h e r e F , F a r e v e c t o r f i e l d s g e n e r a t e d b y к a n d /Г, r e s p e c t i v e l y ( s e e d e f i n i t i o n 1 ) . L e t a , s T e C “ (H , [ 0 , 1 ] ) b e s u c h t h a t

r 0 , р е С A j

a (p ) = 1L 1 . p e A ^

r 0, p e A

a ( P ) = I

a n d a + a > 0 o n H , ___L e t в b e t h e f l o w o f t h e v e c t o r f i e l d a F + a F d e f i n e d o n ÍL ( F i g . 7 ) . It h a s t h e f o l l o w i n g p r o p e r t y :

L e t e Q > 0 b e s o s m a l l t h a t к (ЭГ2, e0) c A 2. F o r e v e r y s ' e B f t t h e c u r v e t -* 0T ( s ) , d e f i n e d f o r n o n - p o s i t i v e t , i n t e r s e c t s t h e s e t к (8Q , e 0) i n e x a c t l y

^ no K ( s , t ) = X ( 7 " 1o 7 ( s )) = k ( y _1o 7 ( s ) , un(t) )

1 , p e C A

X T(P) i f P , x T( p ) e C A

X T(P) i f P , x T( p ) e A 2

3 4 0 KOM ORO W SKI

o n e p o i n t ( b e c a u s e t h e f l o w s в a n d x a r e e q u a l o n A 2 ). T h u s w e c a n d e f i n ea C “ - f u n c t i o n b : 3Q -» I R 1 a s f o l l o w s : i f s e 3 f i , t h e n b ( s ) i s s u c h t h a t

(s) ( T _1° T ( s ) ) ( S i'i, e 0) . W e h a v e f r o m G . 2 ) t h a t b < - | . H e n c e , t h e r e e x i s t C ° ° - f u n c t i o n s v n s u c h t h a t

r “ n ( t ) - l , | s t < f " 1( r )ЭГ2 X ] 0 , f _1( r ) [ Э ( s , t ) -* v n ( s , t ) = j s o m e t h i n g , e0< t < \

b ( s ) - t , t = e0 - t , т е [ 0 , e 0[

a n d 3 v n/ 3 t > 0 .A s t h e n e x t s t e p i n t h e c o n s t r u c t i o n o f фп w e p u t

фпок ( s , t ) = e Vn(s,t) ( 7 _10 7 ( s ) ) (6 )

f o r ( s , t ) e 0 í í X ] e 0/ 2 , f _1( 2 r ) [ , w h a t i s c o m p a t i b l e w i t h d e f i n i t i o n (4 ) .

R e m a r k 5 .

фпOK ( 3a I ) = = х 4 ( э п ) = к ( э п , è)

H . W e i n t r o d u c e t h e f o l l o w i n g m a p s : i f ( s , t ) e 3 f í X ] 0 , f -1 ( 2 r ) [ t h e n % (K ( s , t ) ) : = e Vn(lit) ( ^ o T l s J l e i o S X ] 0 , Г - ! ( 2 г ) [ ) .

O b v i o u s l y , i n t h e l i g h t o f d e f i n i t i o n (6) , w e h a v e фп = ipn o n к ( 3 X ] e 0/2, f ' ^ r ) ! ) . C o m p a r i n g t h e d e f i n i t i o n o f <pn w i t h i t e m C , o n e m a y a s k w h y w e d o n o t s t a t e фп = <pn e v e r y w h e r e . I f t h i s w e r e t h e c a s e , Фп o b t a i n e d i n t h i s w a y w i l l n o t f i t s m o o t h l y Ф п ^ ^ с ) -

R e m a r k 6 . F o r e v e r y 0 ё т < e 0 t h e s e t к (3Q, е 0 - т ) i s m a p p e d b y <pn o n t o i t s e l f . In f a c t , i f w e t a k e к ( s , e 0 - t ) , s e 3 f 2 , t h e n <рп(к ( s , e 0 - t ) ) =

e vn(s ,e 0- T) ( T _ 1 o 7 (s )) = 6b (s) - T(7 " 1o 7 ( s ) ) = 0 . T° 0 b(s) ( 7 ' 1o 7 ( s ) ) =

x - r ° eb(s) (7 ' 1o 7 ( s ) ) e K ( d Q , e0 - t ) .

T h e l a s t e q u a l i t y f o l l o w s f r o m ( 5 ) .

I . F o r e v e r y t e ] 0 , 1 ] t h e m a p 3Í2 3 S ->• к t ( s ) = к ( s , t ) е к ( 3 Q , t ) i s a d i f f e o m o r p h i s m . L e t r e [ 0 , e Q[ ; w e d e f i n e a m a p

d Q 3 s -------- - orT ( s ) = K ¡ l . T ocpn oK4 . T ( s ) e 3 f t

It f o l l o w s f r o m r e m a r k 6 t h a t : 1 ) I m стт = ЭГ2; 2 ) crT d o e s n o t d e p e n d o n n;3 ) i t d o e s n o t d e p e n d o n t , i . e . crT = tr0 f o r r e [ 0 , e 0 [ . T h e e x p l i c i t f o r m u l a f o r aQ i s

3 Q 3 s --------- % ( s ) = K¡ 0l o e b(s) ( 7 ' 1o 7 ( s ) ) e 3 n

I t f o l l o w s f r o m t h e p r o p e r t i e s o f у a n d у a n d f r o m t h e c o n s t r u c t i o n o f t h e f l o w в t h a t ct0 i s a n o r i e n t a t i o n - p r e s e r v i n g d i f f e o m o r p h i s m o f 3£2. S i n c e ЭГ2 i s d i f f e o m o r p h i c w i t h S 2 , ct0 i s s m o o t h l y i s o t o p i c w i t h i d an ( s e e R e f . [ 2] ) . I n o t h e r w o r d s , t h e r e e x i s t s a n C “ - i s o t o p y IR 1 X 3f2 3 ( t , s ) -*■ G t ( s ) £ 3 f 2 s u c h t h a t ,

IAEA-SM R-11/39 341

F i n a l l y w e p u t i¡/n o i c ( s , t ) = Kt o G j O k ”1 о <рпок. ( s , t ) , f o r ( s , t ) e 9 f i X ] 0 , f " \ 2 r ) [ , w h a t i s c o m p a t i b l e w i t h ( 6 ) . S i n c e

r cpn o n к (ЭП X ] f _1( 2 r ) [ )

Фп = ] е 0L i d fi o n к (9 f i X ] 0 , — [ )

t h e m a p Ф п = i~otpnoi~^ g iv e s , w i t h d e f i n e d in B . d i f f e o m o r p h i s m s Ф„| T, a n d Ф J . , a d i f f e o m o r p h i s m f r o m X & C o n t o X & C .

U2 n lli î ( c ) r v v

J . T h u s w e h a v e c o n s t r u c t e d t h e d i f f e o m o r p h i s m s ^ ¡ X & C - ' - X & C . N o w w e a r e g o i n g t o s h o w t h a t t h e s e q u e n c e Фп s a t i s f i e s th e h y p o t h e s i s o f o u r t h e o r e m . I t i s e n o u g h t o p r o v e t h a t i f ш е Г ^ Т * ( X ) ) , t h e n <ï£ u - & n_ ^ ► 0 i n t h e t o p o l o g y o f Г ^ Т * ( X & C ) ) .

1 . W e h a v e f r o m B . a n d D . t h a t Фп = Ф 0 o n U ^. H e n c e , o n U x, Ф,„ to - ¿J =

Ф о 0 ( ^ х ) * ш - (i'x1) ^ = ( i x ) * u - (i'x1) * « = ° -2 . F r o m R e m a r k 2 w e k n o w t h a t

u = 0 o n i n (к (ЭП X ] 0 , j ] ) u C ) = 0

u = 0 o n i~ (к (ЭП X ] 0 , | ] ) U C ) = 0

B u t r e m a r k 5 t e l l s u s t h a t Ф п(9<9) = 9 0 , h e n c e Фп (<9) = &. T h u s =0 - 0 = 0 o n 0 .3 . L e t u s d e n o t e W = i n oK__(8f2 X ] \, f _1( 2 r ) [ ) . S i n c e С ( U i U 0 ) c W , i t i s s u f f i c i e n t to s h o w t h a t ( Ф * и - i5 ) |w p j * 0 in t h e t o p o l o g y o f r k( T * (W )) . B u t

i f i0 K m a p s ЭП X ] 1 , f _1( 2 r ) [ d i f f e o m o r p h i c a l l y o n t o W . S o , w e s h a l l p r o v e t h a t

( i n OK) * ( ^ Ц ~ Ц ) n ^ « , “ 0

i n t h e t o p o l o g y o f Г к( Т * (9 f i X ] j , Г \ 2 г ) [ ) ) .L e t X b e t h e e x p o n e n t i a l m a p

9УГХ ] 0 , 1[ Э ( x , t ) ------► X ( x , t ) = e x p ( t x ) e X

w h e r e . S f i s t h e u n i t b a l l a t 0 i n T XQ ( X ) .F r o m t h e d e f i n i t i o n o f ш a n d ш a n d f r o m t h e f a c t t h a t Фп = i n o n W w e g e t

( i fiO K ) * ( Ф * ш - и ) = ((Х~1о7офпэ к ) * - (А.'1 о у o k ) * ) (X * и)

w h e r e Х~1о7офпок = :a n a n d X_1o i / o k =:a m a p 9f i X ] } , f~\2r)[ d i f f e o m o r p h i c a l l y o n t o t h e i r i m a g e i n Э-5ГХ ] 0 , 1 [ ; h e r e Х * и е Г ” ( Т * ( 3 УГХ ] 0 , 1[ ) ) . B e s i d e s

e n( s . t ) = Í y - ^ T Í S ) . f ° u n( t ) ) = ( 7 _1o 7 ( B ) . F n (t))

a ( s . t ) = Cy ’ ^ Y ( s ), f (t))

I t f o l l o w s f r o m F t h a t an — j - a i n t h e t o p o l o g y o f t h e s p a c e

C k + 1( 9 f i X ] j , f * ( 2 r ) [ , 9X X ] 0 , 1[ ) . H e n c e , w e h a v e o b t a i n e d t h a t ( a * - a * ) ( X * u ) 0 i n t h e t o p o l o g y o f Г к( Т * ( 9 f i X ] i , f - 1( 2 r ) [ )) .

342 KOMOROWSKI

FIG. 8. Proof o f proposition 1.

P r o o f o f p r o p o s i t i o n 1 ( F i g . 8)

W e b e g i n w i t h f e w p r o p e r t i e s o f t h i s d i a g r a m . L e t к ( s . t j e f y ; t h e nç o /u o k ( s , t ) = (f i iexp^. f ( t ) (<p_1) . o 7 ( s ) ) = e x p g <%\(f(t) О?'1). 0y ( s ) ) =e x p g f ( t ) i p ' O Í c p ' 1) , oy (s ) = e x p g f ( t ) y ( s ) = v o k ( s ' , t ) .W e d e f i n e a d i f f e o m o r p h i s m ф : Y & C - ^ X & C a s f o l l o w s :

p u

Г Ф (3Y ( y) ) = i x o<p(y) , y e Y

1 Ф (3Q (P )) = i n (P) , p e i i f

T h i s d e f i n i t i o n i s c o r r e c t b e c a u s e i f j Y(y ) = j ^ Í P ) . t h e n i x °< p (y ) = i n o v ‘ l o V (y ) i j j o y i o i p o j ' ^ o j j j i p ) = i ç j o v ^ o ip o n (p) = i ñ ( p ) . L e t u s n o t i c e t h a t i f w e d e n o t e h== (pf g , t h e n

0 * g = & ( ? )

( s e e E q . ( 1 ) ) . I n d e e d , тг ‘ (<¡p* g ) = ( j ' 1) * qr g = ( . p o f 1) * g = ( i * ^ ) * g =

Ф" ((iy’f g ) = Ф" I-N o w w e p a s s t o o u r p r o o f . L e t u s t a k e U и (V ) e ([ g ] ) = с >2 ([ h ] ),

w h e r e U e -M l g ] ) , V e 3 T ( [ f t ] ) . W e h a v e t o f i n d U ' e .Ж ([ g ] ) a n d V 1 e Л ([ g ] )— t h e n U 1 U ( V 1 n . ) e ^ Sl(f Si ) ~ s u c h t h a t

U ' u ( V ' n ^ ) C U u ( V f l „ # ) ( 8 )

W e s t a t e

U 1 := U (9 )

T h u s w e h a v e o n l y to d e t e r m i n e V 1 . A t t h e b e g i n n i n g o f s e c t i o n 3 w e h a v ei n t r o d u c e d t h e c a n o n i c a l m a p s

IA EA-SM R-11/39 3 4 3

It f o l l o w s f r o m t h e c o n t i n u i t y o f j o i t h a t W = ( j o i ) - 1 (V ) i s a n e i g h b o u r h o o d o f ft i n M ( Y & C ) . S i n c e ф i s a d i f f e o m o r p h i s m , ф* i s a n i s o m o r p h i s m . T h u s , _ m a k i n g u s e o f r e l a t i o n ( 7 ) , w e g e t t h a t (ф'1)* (W) i s a n e i g h b o u r h o o d o f g € M ( X & C ) . H e n c e ^ f r o m r e m a r k 3 , V = [ W ] = [ ft//"1) * (W )] i s a n e i g h b o u r h o o d o f [ g ] e „ < f . S o , t h e r e e x i s t s V 1 e ^V([ g ] ) s u c h t h a t V ' С V .T h i s V 1 a n d U' d e f i n e d in (9) s a t i s f y c o n d i t i o n (8 ).

P r o o f o f p r o p o s i t i o n 2 . L e t u s t a k e U U ( V n „ < f ) e &>s ([ g ] ) , w h e r e U e ^ f j l g ] ) , У е 1 Ж ( [ ё ] ) . W e a r e g o i n g t o f i n d U ' e ([ g ] ) a n d v ’ e J F ([ g ] ) s u c h t h a t

U ' u ( V ' n . . < f ) С U U ( V n ^ ) (10)

W e s t a t eU ' : = U ( U )

T h e r e i s o n l y V 1 t o b e d e t e r m i n e d . T h e m a p s

M ( X & C ) — Î— ►„‘i r ( X & C ) --Í— ► J i 7M M

w e r e i n t r o d u c e d a t t h e b e g i n n i n g o f s e c t i o n 3 . _I t f o l l o w s f r o m t h e i r p r o p e r t i e s t h a t ( j o i )"1 (V ) i s a n e i g h b o u r h o o d o f g e , < f ( X & C ) . 2

In t h e n e x t w e s h a l l d e a l w i t h a n o r m , || ||k, i n t h e s p a c e Г к( ® s T * ( X ) ) ,к = 0 ^ 1 , 2 , . . . , X e E . I f g e M ( X ) , e e E +, t h e n w e d e f i n e 0 ( g , k , e ) : ={ g ' e M ( X ) : I g '-g||k < e } . v

R e t u r n i n g t o o u r p r o o f , s i n c e { 0 { g , k , e ) : k e N , e e I R +} i s a b a s e o f n e i g h b o u r h o o d s o f t h e j>oint g e M ( X & C ) , t h e r e e x i s t k 0 £ N a n d e 0e I R + s u c h t h a t 0 ( § , k 0, e 0) c ( j o i ) ( V ) . I t f o l l o w s f r o m t h e F u n d a m e n t a l T h e o r e mt h a t t h e r e e x i s t s a d i f f e o m o r p h i s m Ф : X & C ^ X & C s u c h t h a t

ft и

« * g - g | | k 0 < Ÿ ( 12 )

I f w e d e n o t e W'-= 0 ( g , k 0, и °м— ), t h e n Ф* (W) С 0 (g , k 0, -£■). T h u s , w eN 0

o b t a i n f r o m ( 1 2 ) t h a t ÿ :‘ ( W ) C 0 ( | , к о , ^ ) . B e s i d e s , s i n c e W i s a n e i g h b o u r h o o d o f g e M ( X fe C ) , i t f o l l o w s f r o m r e m a r k 3 t h a t [ W ] i s a n e i g h b o u r h o o d o f [ g ] e ^ < f . H e n c e t h e r e e x i s t s V 1 e ([ g ] ) s u c h t h a t V 1 С [ W ] . T h i s c h o i c e o f V 1 i s g o o d b e c a u s e V 1 С [ W ] = [ 3 r (W )] С [ 0 ( £ , k 0, e 0)] С V a n d s o d e f i n e d V 1 — w i t h U 1 ( s e e (11 )) — s a t i s f i e s ( 1 0 ) .

R e m a r k 7 . T h e p r e v i o u s r e s u l t s c a n a u t o m a t i c a l l y b e g e n e r a l i z e d f o r t h e c a s e o f n - d i m e n s i o n a l R i e m a n n i a n С - m a n i f o l d s , w h e r e n i s s u c h t h a t e v e r y o r i e n t a t i o n - p r e s e r v i n g d i f f e o m o r p h i s m o f S n_1 i s i s o t o p i c w i t h i d e n t i t y .I f t h i s i s t h e c a s e w e s h a l l o b t a i n t h i s g e n e r a l i z a t i o n b y t h e r e p l a c e m e n t o fS 2 b y S " ’ 1.

5 . T H E N E W T O P O L O G Y IN „ < f A N D T H E A N N I H I L A T I O N O F C H A R G E S- A N O P E N P R O B L E M

T h e e l e c t r i c f i e l d ( o r f l o w o f e l e c t r i c f i e l d , r e s p e c t i v e l y ) g e n e r a t e d b y a p a i r o f o p p o s i t e c h a r g e s c a n b e d e s c r i b e d b y a h a r m o n i c 1 - f o r m ( o r 2 - f o r m ,

3 4 4 KOM OROW SKI

r e s p e c t i v e l y ) o n a t h r e e - d i m e n s i o n a l R i e m a n n i a n C “ - m a n i f o l d w i t h a h a n d l e . T h i s c o n c e p t i s b r i e f l y d e s c r i b e d in R e f . [ 3 ] , c h a p t e r 2 , s e c t i o n 1 1 , a n d , e x t e n s i v e l y , in R e f . [ 4 ] .

A c c o r d i n g t o t h i s i d e a w e h a v e t h e f o l l o w i n g c o r r e s p o n d e n c e s :

p a i r o f c h a r g e s ( + q , - q ) -«--------- •- h a n d l e i n a t h r e e - d i m e n s i o n a li n a u n i v e r s e R i e m a n n i a n m a n i f o l d X # ( S 2 X S 1)

w i t h a m e t r i c g .4f l o w o f e l e c t r i c f i e l d -•--------- ► е е Г ( A T * ( X # ( S 2 X S 1))) s u c h t h a tg e n e r a t e d b y t h i s p a i r 1 . A g e = 0 ( h a r m o n i c i t y ) ,

2 . i f C 2 i s a 2 - c y c l e o n X # ( S 2 X S 1) a r o u n d t h e a d d e d h a n d l e t h e n 4a

3 . i f i s a 2 - c y c l e n o n h o m o l o g o u s t o C 2 t h e n

c2a t t r a c t i o n o f c h a r g e s -«---------► p r o c e s s in w h i c h t h e d i s t a n c e

b e t w e e n c h a r g e s ( ' e n d s ' o f t h e h a n d l e ; s e e f o o t n o t e 4 ) d e c r e a s e s .

T h e s e c o r r e s p o n d e n c e s s u g g e s t t h a t i f o n e w a n t s t o d e s c r i b e th e a n n i h i l a t i o n o f p a i r o f c h a r g e s ( + q , - q ) o n e h a s t o i n t r o d u c e s u c h a t o p o l o g y i n Л f o r w h i c h s u c h a n e v o l u t i o n o f m e t r i c s ( c o n t i n u o u s c u r v e i n Л ) w o u l d b e p o s s i b l e t h a t l e a d s t o v a n i s h i n g o f t h e h a n d l e . M o r e o v e r , o n a c c o u n t o f t h e a t t r a c t i o n p r o c e s s , t h i s v a n i s h i n g m u s t c o n s i s t in t h e d i s a p p e a r a n c e o f t h e h a n d l e , b u t n o t in i t s b r e a k i n g ( F i g . 9 ) , a s w a s s u g g e s t e d i n R e f . [ 5 ] a n d b y s e v e r a l o f W h e e l e r ' s f i g u r e s i n R e f . [ 3 ] .

S u c h a n e w t o p o l o g y in „4? w a s i n t r o d u c e d a n d i t s c a n o n i c a l c h a r a c t e r w a s p r o v e d i n t h e o r e m _ 2 . T h e b a s i c i d e a c o n s i s t s i n t h e i d e n t i f i c a t i o n o f e a c h g e M ( X ) w i t h g e M ( x & C ) ? M ( X # ( S 2 x s 1)).

T h u s , a n a r e n a w h e r e t h e a n n i h i l a t i o n o f c h a r g e s m a y t a k e p l a c e , i s p a r t i a l l y p r e p a r e d . N o w , i f w e w a n t t o d e s c r i b e t h e a n n i h i l a t i o n , w e h a v e t o b e a b l e t o p a s s c o n t i n u o u s l y f r o m h a r m o n i c 2 - f o r m s o n X § ( S 2 X S 1) S ' X & С t o h a r m o n i c 2 - f o r m s o n X .

O n e c a n t r y t o d o t h a t b y a n a n a l o g o u s i d e n t i f i c a t i o n o f e v e r y e e Г ( Л Т * ( Х ) ) w i t h ё e Г ( A T * ( X & С ) ) ~ Г (Â T * (X # ( S 2 X S 1) ) ) . L e t u s t a k e a c u r v e

] - 1 , 1 [ B t -------- ► ( g t , e t )

4 A physicist may demand the Riemannian manifold (X$(S2 x S1), g) to have the handle. S2 x S1, with a 'very small radius1 added, i.e . inf HC2 « radius o f electron, where II II g is the length in the sense of the metric g and inf is taken over all 2-cycles in X $ (S2 x S1) that go around the added handle. Then a concept o f localization o f charges can be introduced because not the whole o f the handle is accessible to our physical (metrical) experiments — we are larger than the electron!

4a Since now we write Г instead of Г 00, however, we shall only deal with C°°-objects.

IA EA-SM R-11/39 3 4 5

FIG. 9. Disappearance and breaking of a handle

w h e r e

( g j , e t ) £ M ( X & С ) X Г ( A T * ( X & C ) ) f o r t < 0

( g t , e ) e M ( X ) X Г ( Л Т * ( X ) ) f o r t ё 0

a n d

( g t . e ) e 0) e M ( X & C ) X Г ( А т * ( Х & С ) )

B y t h e a b o v e i d e n t i f i c a t i o n o u r c u r v e i s c o n t i n u o u s . L e t C 2 b e a 2 - c y c l e o n X & C g o i n g a r o u n d t h e h a n d l e a d d e d to X . W e a s s u m e t h a t A Rt e t = 0 f o r a l l t a n d t h a t

w e h a v e t h a t

H e n c e , w e g e t a c o n t r a d i c t i o n b e c a u s e , a c c o r d i n g t o r e m a r k 2 , w e c a n c h o o s e s u c h C 2 t h a t ё„ = 0 o n C 2 . T h i s p r o v e s t h a t t h e p r o p o s e d i d e n t i f i c a t i o n o f e a n d e — t h e t r i c k u s e d i n t h e c o n s t r u c t i o n o f t h e n e w t o p o l o g y i n -4? — d o e s n o t l e a d t o s u c h t o p o l o g y in

U Г ( Л Т * ( X ) )

Х Е ЕD i f f

t h a t w o u l d b e g o o d f o r a d e s c r i p t i o n o f t h e a n n i h i l a t i o n o f c h a r g e s . ( H e r e E i s t h e s e t o f a l l t h r e e - d i m e n s i o n a l C “ - m a n i f o l d s . )

H o w d o e s t h e f l o w o f e l e c t r i c f i e l d c h a n g e w h e n t w o o p p o s i t e c h a r g e s a r e c o m i n g c l o s e r a n d c l o s e r ? P h y s i c a l e x p e r i m e n t s t e l l u s t h a t t h e f l o wa ) i n c r e a s e s b e t w e e n c h a r g e s ;b ) d e c r e a s e s t o z e r o a t p o i n t s f a r , i n c o m p a r i s o n w i t h t h e d i s t a n c e o f t h e

c h a r g e s , f r o m t h e c h a r g e s ;c ) t h e t o t a l f l o w r e m a i n s c o n s t a n t .

J e t I = 1 f o r t < 0 (u n i t c h a r g e s ) . T h e n , f r o m c o n t i n u i t y ,c 2

/ l i m e t = l i m / e t = 1J t-» о t -> о Jc2 c 2

3 4 6 KOMOROWSKI

FIG. 10. Open set U containing the handle.

T h u s , w e m a y s u p p o s e t h a t i f w e h a v e a h a r m o n i c 2 - f o r m e o n a t h r e e - d i m e n s i o n a l m a n i f o l d X ( w i th a h a n d l e ) w i t h a R i e m a n n i a n s t r u c t u r e g a n d e h a s t h e o n l y n o n - z e r o p e r i o d s o n 2 - c y c l e s g o i n g a r o u n d t h e h a n d l e , t h e n w e c a n c h o o s e a n o p e n s e t U c X c o n t a i n i n g t h e h a n d l e 5 ( F i g . 10) a n d a c o n t i n u o u s c u r v e (0 , 1] Э е - ( g e, e e) € M ( X ) X Г ( À T * ( X ) ) s u c h t h a t ( g 1( e j ) =( g , e ) , f o r e v e r y е е ( 0 , 1 ] e £ = 0 a n d t h e p e r i o d s o f e e a r e t h e s a m e a s t h o s e o f e , e

W l i m i , , 0

e _ := l i m e = 06 -> о 6 x \ u

A n e x a m p l e o f s u c h c u r v e e -* ( g £ , e e ) w i l l b e g i v e n l a t e r . T h u s , r e t u r n i n g t o t h e p r o b l e m o f a t o p o l o g y in

2U Г ( Л Т * (X ) )

X £ E _______________D i f f

w e m a y f o r m u l a t e t h e f o l l o w i n g r e m a r k :2 ,

R e m a r k . I t s e e m s t h a t o n e h a s to i d e n t i f y e v e r y е е Г ( Л Т * ( X ) ) w i t h a s e t ë + E „ C Г ( А т * ( X & C ) ) Г ( Л Т ( X # ( S 2 X S 1) ) ) , w h e r e E „ c o n s i s t s o f a l l s e c t i o n s w i t h s u p p o r t o n t h e a d d e d h a n d l e . P r e c i s e l y s p e a k i n g E „ : =| е е Г ( Л Т $ ( X & C ) ) ^ u p p e C f l - ÿ 1 ( x 0) } / c f . p . 5 / . г ________ L e t u s g i v e a f e w d e f i n i t i o n s : F 2 ( X ) : = M ( X ) X Г (А т * ( X ) ) , F 2 ( X ) : =M ( X ) X Г (Л Т * ( Х ) ) , F z:= u F 2( X ) , F ^ ; = U F 2 ( X ) . T h e i s o m e t r y r e l a t i o n

x e c X £ L

i n U M ( X ) ( U M ( X ) , r e s p e c t i v e l y ) i n d u c e s a n e q u i v a l e n c e r e l a t i o n i n F 2 хег XÊE

( o r F 2 , r e s p e c t i v e l y ) , a s f o l l o w s : i f ( g ¡ , e ; ) e F 2( X i ) ( o r F 2 ( X ¡ ) , r e s p e c t i v e l y ) , i = 1 , 2 , t h e n ( g j , e j ) ~ ( g 2, e2) i f a n d o n l y i f t h e r e e x i s t s a d i f f e o m o r p h i s m <p: X j ->• X g s u c h t h a t g j = g 2 a n d e j = ip* e 2 . W e d e n o t e !?'■= F 2/ ~ ,F 2/ ~ . J *-2 ( o r ¿ F 2, r e s p e c t i v e l y ) i s a f i b r e s p a c e o v e r ^ ( o r r e s p e c t i v e l y ) w i t h t h e c a n o n i c a l p r o j e c t i o n ; w e s h a l l s p e a k l a t e r a b o u t t o p o l o g i e s .

5 We say that U contains the handle i f the quotient map X -» X/U kills the homology elements generatedby the handle.

IA EA -SM R -11/39 3 4 7

It f o l l o w s f r o m t h e p r e v i o u s c o n s i d e r a t i o n s t h a t w e a r e g o i n g to l o o k f o r d e s c r i p t i o n s o f a n n i h i l a t i o n s a m o n g l i f t s , t o ip'2, o f c u r v e s i n „ ¿ f ;

h e r e , Л i s e n d o w e d w i t h t h e n e w t o p o l o g y . T h u s t h e q u o t i e n t t o p o l o g y in S ' 2 i s t o o s t r o n g f o r o u r p r o g r a m m e . W e c a n d e f i n e t h e w e a k e r o n e a s it w a s s u g g e s t e d i n t h e a b o v e r e m a r k . T h e i d e n t i f i c a t i o n o f e v e r y ( g , e ) e F 2 ( X ) w i t h { ( g , e + e ) : e e E „ } C F 2 (X & C ) i n d u c e s a ' c o n n e c t e d ' t o p o l o g y i n F 2 (X ) U F 2 (X & C ) . A s t h e w e a k e r t o p o l o g y in .S'-2, w e t a k e t h a t i n d u c e d b y t h e c a n o n i c a l p r o j e c t i o n s :

w h e r e F 2 ( X ) u F 2 ( X & C ) i s e n d o w e d w i t h t h e a b o v e d e f i n e d ' c o n n e c t e d ' t o p o l o g y . ( T h e c o r r e c t n e s s o f t h i s d e f i n i t i o n o f t h e t o p o l o g y i n ^ — i t s i n d e p e n d e n c e o f a c h o i c e o f v 's — c a n b e p r o v e d a s i n t h e o r e m 1 .)

E v e n w h e n J ^2 i s e n d o w e d w i t h t h i s w e a k e r t o p o l o g y w e s t i l l d o n o t k n o w w h e t h e r i t i s a p r o p e r a r e n a f o r t h e d e s c r i p t i o n o f a n n i h i l a t i o n p r o c e s s e s . W e h a v e t o g i v e , a t l e a s t , o n e e x a m p l e o f a n n i h i l a t i o n a s a l i f t , t o S ' -2, o f a c u r v e i n ЛС. (W e n e e d n o t t a k e a c c o u n t o f ' e q u a t i o n s o f m o t i o n ' o f s u c h p r o c e s s e s s i n c e o n l y k i n e m a t i c s i s c o n s i d e r e d . ) T h i s l e a d s u s t o a p r o b l e m w h o s e s o l u t i o n i s u n k n o w n t o t h e a u t h o r .

O n a c c o u n t o f H o d g e ' s , K o d a i r a ' s a n d d e R h a m ' s t h e o r e m s in t h e t h e o r y o f h a r m o n i c f o r m s , i t w i l l b e e a s i e r t o d e a l o n l y w i t h c o m p a c t m a n i f o l d s , i . e . w e t a k e E a s t h e s e t o f a l l c o m p a c t t h r e e - d i m e n s i o n a l C " - m a n i f o l d s .

I f X i s a c o m p a c t 3 - d i m e n s i o n a l C “ - m a n i f o l d w i t h a h a n d l e t h e n w e h a v e t h e m a p 6 M ( X ) B g -> e ( g ) e T ( Á T * ( X ) ) , w h e r e e ( g ) i s t h e u n i q u e 2 - f o r m s u c h t h a t A ge ( g ) = 0 a n d f o r e v e r y 2 - c y c l e C 2

__ L e t b e g i v e n a n o p e n s e t U C X c o n t a i n i n g t h e h a n d l e ; w e d e f i n eМ ц (X)-'= I g e M ( X ) : g = o ) . L e t Р ц Ь е t h e s e t o f a l l c o n t i n u o u s c u r v e s

(0 , 1 ] Э e -2 -*- g e e M ( X ) s u c h t h a t l i m g ee M u ( X ) . N o w w e c a n f o r m u l a t ee-> о

O p e n P r o b l e m 7

I . H o w c a n b e c h a r a c t e r i z e d t h e e l e m e n t s g e P j j f o r w h i c h

II . L e t X - Y & С a n d U = X \ I m i y ( s e e p r e v i o u s l y ) , h e M ( Y ) . D o e s t h e r e e x i s t a n e l e m e n t g e P y s u c h t h a t

F 2 ( X ) U F 2 ( X & C )V

{1 i f C 2 i s g o i n g a r o u n d t h e h a n d l e0 o t h e r w i s e

( * ) s u p p l i m e ( g ) C U ?

V

l i m g £ = h e M y j X ) ( s e e p r e v i o u s l y )

a n d ( * ) i s f u l f i l l e d ?

6 It was shown in Ref. [ 6] that this map (generalized for arbitrary dimensions o f X) is o f C1-class in the sense o f Ref. [7 ] and, o f course, in the sense o f [ 8] .

7 This problem, after straightforward reformulation, can be stated for an n-dimensional compact C°°-manifold X, g€ G M (X ), e (g e) — a harmonic к-form, on X, with prescribed periods, U — an open set in X such that the quotient map X -* X/U shrinks all к-cycles on which the periods o f e (g£) do not vanish.

3 4 8 KOMOROWSKI

T h e f o l l o w i n g t w o e x a m p l e s m a y b e u s e f u l f o r b e t t e r u n d e r s t a n d i n g o f o u r p r o b l e m . L e t g b e t h e c a n o n i c a l R i e m a n n i a n m e t r i c o n S 2C 1R 3 a n d d t ® d t b e t h e c a n o n i c a l R i e m a n n i a n m e t r i c o n S 1 = И 1/ Х . L e t X = S 2 X S 1 a n d U , U j , U 2 b e a s s h o w n i n F i g . 1 1 . I n t h e n e x t w e s h a l l u s e t h e s a m e s y m b o l s t o d e n o t e f o r m s o n S 2 o r S 1 a n d t h e f o r m s o n X i n d u c e d b y t h e m .

E x a m p l e 1 .

L e t g £ = r £ g + ( r £)2 d t ® d t e M ( X ) , w h e r e e e ( 0 , 1 j a n d r £ e C ° ° ( X ) i s s u c h t h a t

f £ f o r X e u r (x ) H TT € s r 5 1 o n U ,

e ' ' [_ 1 f o r X e U 2, e 1

a n d l i m г б С ^ Х ) , T h u s t h e c u r v e e -* g e b e l o n g s t o Р ц .e o 3

L e t u s t a k e e = — * g l g2 , w h e r e i s t h e H o d g e o p e r a t o r o n S 2 i n d u c e d b y

t h e m e t r i c g . It i s e a s y t o c a l c u l a t e t h a t * e = -Д- d t . T h u s e i s a h a r m o n i c & J ge 47Г

2 - f o r m o n X a n d / e = 1 . S o e ( g ) = e f o r e v e r y e e ( 0 , 1 ] . B u t s u p ps2

l i m e ( g ) = s u p p e = X , h e n c e (v ) i s n o t f u l f i l l e d . e ->0 — --------------------------

E x a m p l e 2 .

L e t A æ S 1, i = 1 , 2 , b e o p e n s e t s s u c h t h a t U A ¡ X S J C U . L e t <p‘ e C ~ ( S ' ) ,i ^ J . e

е е (0 , 1 ] , b e s u c h t h a t : 1) 0 < <p* s К f o r e v e r y e e (0 , 1 ] , 2 ) cp‘ = e£ 6 S ^ A j

3 ) l i m / e C " ( S ' ) . M o r e o v e r , l e t с - * 0 e

s2W e d e f i n e g £ = r £ ip2g + ( r dt ® d t , w h e r e r e C " ( X ) i s s u c h t h a t

. f e2 f o r x e U „ 1 TTr . ( x ) = i i , _ TT e ¿ á r S — o n Ц a n d l i m г , с £ С ( X ) .

е ' ' {_ J f o r x e u 2, e 1 о

T h u s t h e c u r v e e -* g e b e l o n g s to P 0 . L e t u s t a k e e £ = <p2 * g l s2 , t h e n w eg e t * e e = <p\d t . T h u s e e i s a h a r m o n i c 2 - f o r m o n X a n d o n a c c o u n t o f

( * * ) w e g e t t h a t e ( g c) = e £. I t i s e a s i l y s e e n t h a t s u p p l i m e f C A j X S 1 C U .€ “** 0

Hence the condition (*) i s fulfil led .

IAEA-SM R-11/39 349

In t h e p r e s e n t s e c t i o n E w i l l b e t h e s e t o f a l l t h r e e - d i m e n s i o n a l o r i e n t e d C “ - m a n i f o l d s a n d b y d i f f e o m o r p h i s m w e s h a l l m e a n a n o r i e n t a t i o n - p r e s e r v i n g d i f f e o m o r p h i s m ; w e s h a l l d e n o t e d t h e m b y D i f f Q.

L e t X e E , g e M ( X ) t h e n t h e s e t S ( g ) o f a l l s p i n - s t r u c t u r e s o n t h e o r i e n t e d R i e m a n n i a n m a n i f o l d ( X , g ) i s d e f i n e d a s t h e s e t o f t h o s e e l e m e n t s o f H 1( F ( g ) , Z 2) w h i c h , r e s t r i c t e d to a n y f i b r e F ( g ) x , x e X , g i v e a g e n e r a t o r o f t h e c y c l i c g r o u p H ^ F f g ) , ; , Z g ) 8; s e e R e f . [ 9] . H e r e F ( g ) i s t h e p r i n c i p a l b u n d l e o f o r i e n t e d g - o r t h o n o r m a l f r a m e s t a n g e n t t o X ; h e n c e H 1 ( F ( g ) x , Z 2 ) = H 1 ( S O ( 3 ) , Z 2) = Z 2 . (It i s o b v i o u s t h a t f o r e v e r y X e E , g e M ( X ) t h e s e t S ( g ) i s n o n - e m p t y ; s e e f o o t n o t e 8 .)

I f g , h e M ( X ) t h e n t h e S c h m i d t o r t h o n o r m a l i z a t i o n i s a n i s o m o r p h i s m F ( g ) -» F ( h ) w h i c h i n d u c e s a b i j e c t i o n S ( g ) ->• S ( h ) . I f w e t a k e g 0e M ( X ) t h e n w e h a v e a b i j e c t i o n M ( X ) X S ( g Q) -» U S ( g ) . U s i n g t h i s b i j e c t i o n w e

g £ M (X )

c a n i n t r o d u c e a t o p o l o g y in th e s e t и S ( g ) . T h i s t o p o l o g y d o e s n o t d e p e n dg e m (X)

o n t h e c h o i c e o f g Q. I n t h e n e x t , w e s h a l l d e n o t e b y S ( X ) ( o r F x , r e s p e c t i v e l y ) t h e s e t S ( g 0) ( o r t h e b u n d l e F ( g 0 ) r e s p e c t i v e l y ) , w h e r e g Q i s a n y c h o s e n e l e m e n t o f M ( X ) . B y t h e s e t o f a l l s p i n - s t r u c t u r e s o n a n o r i e n t e d m a n i f o l d X e E w e s h a l l m e a n t h e t o p o l o g i c a l s p a c e M ( X ) X S ( X ) , w h e r e S ( X ) i s a d i s c r e t e s p a c e .

L e t X , Y e E a n d <p : X -*■ Y b e a n ( o r i e n t a t i o n - p r e s e r v i n g ) d i f f e o m o r p h i s m1 i n t o 1 , t h e n tp i n d u c e s a s m o o t h b u n d l e i n j e c t i o n F ^ : F x -*■ F Y . H e n c e F * : H 1 ( F y , Z 2) -► H ^ F j j , Z 2) g i v e s t h e m a p F * : S ( Y ) -* S ( X ) . In p a r t i c u l a r , w e h a v e d e f i n e d a m a p o f D i f f 0( X , Y ) i n t o b i j e ' c t i o n s f r o m S ( Y ) t o S ( X ) . T h e a i m o f t h i s s e c t i o n i s to i n t r o d u c e s u c h a t o p o l o g y i n t h e s e t

U M (X ) X S (X )

___________D i f f 0

w h i c h w o u l d h a v e p r o p e r t i e s a n a l o g o u s t o t h o s e o f t h e ' n e w 1 t o p o l o g y in L e t u s b e g i n w i t h t w o l e m m a s :

L e m m a 2 . I f U i s a c o n t r a c t i b l e s u b s e t o f X e E s u c h t h a t X \ U i s a n o p e n s u b m a n i f o l d , t h e n t h e c a n o n i c a l m a p F r : S ( X ) -» S ( X \ U ) , w h e r e i : X \ U -* X , i s b i j e c t i v e .

P r o o f . I t s u f f i c e s to c o m p a r e a q u a n t i t y o f s p i n - s t r u c t u r e s o n X a n d o n X \ U ; s e e R e f . [ 9 ] .

L e m m a 3 . I f U , V a r e o p e n s u b m a n i f o l d s o f X e E s u c h t h a t U D V i s c o n n e c t e d a n d H 1 ( U n V , Z 2) = 0 t h e n t h e c a n o n i c a l h o m o m o r p h i s m X = F f X F f ,

X : h ! ( F x , Z 2) — * H ^ F j j . Z 2) © H ! ( F V, Z 2)

w h e r e i v ; U -*■ X , i y : V -*■ X , g i v e s a b i j e c t i o n X : S ( X ) -»■ S (U ) X S (V )

6. CONTINUOUS CHANGES O F SPIN -ST R U C T U R ES

8 It is known [10 ] that i f X is a three-dimensional oriented manifold then T(X) is a trivial bundle. This fact w ill not be used except in remark 4' ,2). The author wants to proceed a way which would be useful in generalizations for dimensions93.

3 5 0 KOM OROW SKI

P r o o f . L e t u s t a k e t h e f o l l o w i n g d i a g r a m

Jt f ju n v , Z.)

I0 - h ! ( F x , z 2) - H i ( F U( Z „ ) Ф H 1 ( F , Z . ) 4 H 1( F П F , Z J = H 1( F , Z J

Г'

Н ^ Б О С З ) , Z 2)

w h e r e t h e h o r i z o n t a l s e q u e n c e i s a p a r t o f t h e M a y e r - V i e t o r i s e x a c t s e q u e n c e a n d t h e v e r t i c a l o n e i s a n e x a c t s e q u e n c e e x t r a c t e d f r o m t h e s p e c t r a l s e q u e n c e o f t h e f i b r a t i o n F u n v - O b v i o u s l y , X ( S ( X ) ) C S ( U ) X S ( V ) .

O n t h e o t h e r h a n d , i f (cr, a ' ) e S ( U ) X S ( V ) t h e n i ,;‘ o/u (cr, a ' ) = 0 a n d s i n c e i * i s i n j e c t i v e w e g e t t h a t ц. (cr, cr ') = 0 . H e n c e t h e r e e x i s t s p e H 1 ( F x , Z 2 ) s u c h t h a t X(p) = ( a , cr ' ) . B u t a , cr' a r e s p i n - s t r u c t u r e s , t h u s p m u s t b e l o n g to S ( X ) .

L e t у b e a p o i n t a d d i t i o n o f С t o X e E a t a p o i n t x 0e X ( s e e s e c t i o n 1 ) .I t c a n b e s e e n t h a t a n o r i e n t a t i o n o f t h e m a n i f o l d X & С i s d e t e r m i n e d b y t h e o r i e n t a t i o n o f X . T h u s X ^ C e E . v

W e a r e g o i n g t o d e f i n e a s u r j e c t i o n

a, , : S ( X & C ) -------- ► S ( X )V

L e t ( ( f i , к ) , g , S>f , y , f } b e t h e p a r a m e t e r s o f v . W e d e f i n e Vr = î r ÿ ^ X X e x p Х г ) С л & C , w h e r e 0 < r 5 p f , t h e n 7r „ | V r i s a d i f f e o m o r p h i s m o n t o i t s i m a g e . L e t U = I m i Q С X & С , t h e n Vr U U = X & С a n d Vr П U i s d i f f e o m o r p h i c t o S 2 X I . L e t u s t a k e t h e f o l l o w i n g c o m m u t a t i v e d i a g r a m

S ( X & C ) — S (U ) X S (Vr )

w h e r e X i s t h e b i j e c t i o n d e f i n e d i n L e m m a 3, s e c o n d f a c t o r , j : V X & С i s t h e n a t u r a l i n j e c t i o n a n d

■ X \ e x p S T r — =— ► X

B y L e m m a 2 t h e m a p F . ,,! i s a b i j e c t i o n . T h e a b o v e d i a g r a m d e f i n e s aio TTys u r j e c t i o n a v : S (X & C ) -* S ( X ) . I t c a n b e s e e n t h a t a u d o e s n o t d e p e n d o n r a p p e a r i n g i n t h e d e f i n i t i o n . ( L e t u s n o t e t h a t U w a s n o t u s e d in t h e a b o v e d e f i n i t i o n o f au. )

P r o p o s i t i o n 3 . I f c r e S ( X ) t h e n a 1 (cr) i s a 2 - e l e m e n t s e t .

P r o o f . It f o l l o w s f r o m t h e a b o v e d i a g r a m t h a t t h e n u m b e r o f e l e m e n t s in <*"1( c r ) i s e q u a l t o t h e n u m b e r o f e l e m e n t s i n S ( U ) . U s i n g t h e L e m m a in R e f . [ 9 ] w e o b t a i n t h a t i t i s e q u a l t o t h e n u m b e r o f e l e m e n t s i n H 1( U , Z 2) ^ H ^ S 2 x S 1, Z 2) ^ Z 2.

IA EA-SM R-11/39 351

T h e s u r j e c t i o n a u w a s d e f i n e d f o r a c h o s e n p o i n t a d d i t i o n v . I t s i n d e p e n d e n c e o f t h i s c h o i c e m a y b e e x p r e s s e d a s

T h e o r e m 4 . L e t X e r b e e n d o w e d w i t h a R i e m a n n i a n m e t r i c g e M ( X ) . I f v, a r e p o i n t a d d i t i o n s o f С t o X a t x 0€ X , и € Г " ( Х T * ( X ) ) a n d k e N t h e n t h e r e e x i s t s a s e q u e n c e o f ( o r i e n t a t i o n - p r e s e r v i n g ) d i f f e o m o r p h i s m s

Ф : X & С -------- - X & Сn и ¡r

s u c h t h a tФ * u - u ---------- »■ 0П —> oo

i n t h e t o p o l o g y o f Г к ( ® T * ( X ) ) a n d f o r e v e r y n t h e d i a g r a m

S ( X )

i s c o m m u t a t i v e .

P r o o f . A s t h e d i f f e o m o r p h i s m s Фп w e s h a l l t a k e t h o s e c o n s t r u c t e d i n t h e p r o o f o f t h e f u n d a m e n t a l t h e o r e m . I t i s e a s y to s e e t h a t t h e y p r e s e r v e t h e o r i e n t a t i o n . T o p r o v e t h e c o m m u t a t i v i t y o f o u r d i a g r a m , l e t u s d e f i n e a ( o r a~, r e s p e c t i v e l y ) u s i n g Vt = w'J ( X \ e x p 5Tr ) ( o r \^r = 7r~ ( X \ e x p r e s p e c t i v e l y ) , w h e r e r = \ m i n ( p f , pj"). In s e c t i o n 4 B , t h e s e t Vr w a s d e n o t e d b y U j . L e t u s n o t i c e t h a t ФП(УГ ) = V r a n d t h e d i a g r a m

X

i s c o m m u t a t i v e . T h u s w e g e t c o m m u t a t i v i t y o f

S ( X )

w h i c h c o m p l e t e s t h e p r o o f .N o w w e c a n p a s s to t h e c o n s t r u c t i o n o f a t o p o l o g y in . L e t

( g , f f ) C M ( X ) X S ( X ) ; b y [ g , a ] w e d e n o t e t h e c l a s s in . c o n t a i n i n g ( g , ст).W e s h a l l u s e a l s o t h e f o l l o w i n g n o t a t i o n : i f © C M ( X ) X S ( X ) ( o r& С M ( X ) X S ( X ) ) , r e s p e c t i v e l y ) t h e n [ & ] i s t h e s e t o f a l l c l a s s e s i n ,0C

U M ( X ) X S ( X ) s. X € E

------------— — ------------, r e s p e c t i v e l y ) c o n t a i n i n g t h e e l e m e n t s o f О .s ¿Л П л

3 5 2 KOMOROWSKI

L e t s - ( X , g , x , v ) b e a s i n s e c t i o n 3 . W e d e f i n e t h e f o l l o w i n g f a m i l y o f s u b s e t s i n л ? :

S

^ ( [ g , d ) = | [ D X { c r } ] u ( [ V X { с т ' , а " } [ n „<*;) : U e . - f ( g ) , V e Щ

w h e r e - ^ ( g ) ( o r JV ^ ^ r e s p e c t i v e l y ) i s a b a s e o f n e i g h b o u r h o o d s o f t h e p o i n t g e M ( X ) ( o r g e M ( X ) , r e s p e c t i v e l y ) a n d { c t ' . c t " } = a'J- (ct); s e e p r o p o s i t i o n 3 .

F o r s o d e f i n e d ([ g , c r ] ) w e c a n g e t a n a n a l o g o f t h e o r e m 1 . I n i t s p r o o f w e h a v e t o m a k e u s e o f t h e o r e m 4 i n t h e p l a c e o f t h e f u n d a m e n t a l t h e o r e m .

P r o c e e d i n g a s i n s e c t i o n 2 , w e t a k e ^ " a n d IN, w h e r e E c o n s i s t s o f o r i e n t e d m a n i f o l d s , a n d v i a w e d e f i n e O n t h e b a s i s o f t h ea b o v e m e n t i o n e d a n a l o g u e o f t h e o r e m 1 , w e c a n o b t a i n t h e o r e m 2 in i t s s p i n - s t r u c t u r e v e r s i o n , i . e . f o r t h e a b o v e d e f i n e d t o p o l o g y i n , < i \

N e x t w e g e t t h e f o l l o w i n g v e r s i o n o f t h e o r e m 3 .

T h e o r e m 3 ' . T h e t o p o l o g y i s w e a k e r t h a n t h e n a t u r a l ( q u o t i e n t )t o p o l o g y i n H o w e v e r , i f X e E t h e n b o t h t o p o l o g i e s r e s t r i c t e d to[ M ( X ) X S ( X ) ] C „ < f s c o i n c i d e .

A ' c o n n e c t e d n e s s 1 o f t h e j u s t i n t r o d u c e d t o p o l o g y in c a n b ee x p r e s s e d a s

R e m a r k 4 ' .

1 ) I f X e E t h e n X ( p ) = X # ( S 2 X S 1) # . . . # ( S 2 X S ^ G E a n d U [ M ( X ( p ) ) Xp times p = 0

S ( X ( p ) ) ] c , ^ i s c o n n e c t e d w h e n , ^ i s e n d o w e d w i t h t h e t o p o l o g y - ^ IN*

2 ) L e t ^ Z ( S 3(p ) ) = ^ ( S .3( P ^ - S l S- i ( E i l a n d . ^ 0( S 3 (p ) ) = ] • T h e n t h e

c a n o n i c a l p r o j e c t i o n ( S 3 (p ) ) -* a ( P + l ) - f o l d c o v e r i n g .

P r o o f . 1 ) f o l l o w s d i r e c t l y f r o m t h e d e f i n i t i o n o f :7¡jrjN a d 2 ). L e t X e E t h e n F x = X X S O ( 3 ) ( s e e f o o t n o t e 8 ) . L e t ct0 b e t h e g e n e r a t o r o f H 1 ( S O ( 3 ) , Z 2).W e s h a l l d e n o t e a l s o b y a 0 t h e i n d u c e d e l e m e n t o f H X( F X , Z 2) . T h u s a 0e S ( X ) . I t i s s e e n t h a t S ( X ) ={ста = ст0 + т * » : a e H ^ X , Z 2) } , w h e r e тг : F x -* X . L e t u s n o t e t h a t i f ^ e D i f f 0( X ) t h e n ( F * c t q, = a g ) « = ► (q?fa = /3). N o w w e t a k e X = S 3 ( p ) . L e t t t = ( « 1, - . . . O p ) . (3 = ( Эр . . . , | З р ) е ф Z 2 = H i ( S 3 ( p ) , Z 2) . It c a n b e p r o v e d t h a t

( H < p e D i f f 0 S 3(p) : 3 = Г * ) < « = ► ( H e r e n ( p ) : ¡3. = * „ , „ ) * ( ” )

T h i s f o l l o w s f r o m t h e f a c t t h a t i f U x, . . . , Up a r e ( s m a l l ) d i s j o i n t b a l l s o n S 3 t h e n f o r e v e r y а е п ( р ) t h e r e e x i s t s a n o r i e n t a t i o n - p r e s e r v i n g d i f f e o m o r p h i s m cp s u c h t h a t <p(U¡) = U 0 (¡) . W e g e t f r o m (*) t h a t

H 1( S 3(p ) ) = ® z 2 D i f f 0 П (p)

w h e r e o n t h e r i g h t - h a n d s i d e w e h a v e a s e t o f p + 1 e l e m e n t s .R e m a r k 4 ' i s i l l u s t r a t e d b y F i g . 1 2 .

9 n ip ) is th e group o f p e rm u ta tions o f p e lem en ts .

IAEA-SM R-11/39

1 1

FIG. 12. Diagram illustrating remark 4 '.


I s h o u l d l i k e t o e x p r e s s m y g r a t i t u d e t o P r o f e s s o r K . M a u r i n f o r h i s i n t e r e s t i n t h i s w o r k a n d t o D r . W . S z c z y r b a f o r a n e l e g a n t p r o o f o f l e m m a 1 .

R E F E R E N C E S

[1 ] FISCHER, A.E. , The Theory of Superspace, Relativity, Plenum Press, (1970) p. 303.[2 ] SMALE, S . , Diffeomorphisms of the 2-sphere, Proc. Am. Math. Soc. 10 (1959) 621.[3 ] WHEELER, J .A . , Einstein's Vision, Springer-Verlag, Berlin (1968).[4 ] WHEELER, J .A . , Geometrodynamics, Academic Press, N .Y . (1962).[5 ] WHEELER, J.A . , Particles and Geometry, Relativity, Plenum Press (1970) 31.[6 ] SZCZYRBA, W. , Harmonic forms and deformations of metrics on Riemannian manifolds (unpublished).[7 ] SZCZYRBA, W. , Differentiation in locally convex spaces, Studia Math. 39 (1971) 291.[8 ] OMORI, H. , On the group of diffeomorphisms on a compact manifold, Proc. 1968 AMS Summer Inst,

on Global Analysis_15 (1968) 167.[9 ] MILNOR, J. , Spin structures on manifolds, L'Enseignement mathématique (1962) 198.

[10] STEENROD, N ., The Topology o f Fibre Bundles, Princeton Univ. Press (1951).

IAEA-SM R-11/40

A NEW PROOF FOR REGULARITY OF SOLUTIONS OF ELLIPTIC DIFFERENTIAL OPERATORS

M. KURANISffl D epartm ent of M athem atics,Colum bia University,New York, N. Y . ,United States of Am erica

Abstract

A NEW PROOF FOR REGULARITY OF SOLUTIONS OF ELLIPTIC DIFFERENTIAL OPERATORS.A proof, which avoids use of the mollifier and, instead, employs the power of the Laplacians and the

notion of index, is given for the regularity of the solutions of elliptic differential operators.

1 . L e t L b e a p a r t i a l d i f f e r e n t i a l o p e r a t o r . A s s u m e t h a t L u = v , v i s o f c l a s s C ~ , a n d t h a t u i s o f c l a s s C k. I f L i s e l l i p t i c a n d c o e f f i c i e n t s a r e C ° ° , i t f o l l o w s t h a t u i s o f c l a s s C ° ° . U s u a l l y , w e p r o v e t h i s b y m e a n s o f S o b o l e v n o r m s a n d t h e m o l l i f i e r , o r b y c o n s t r u c t i n g a p a r a m a t r i x b y m e a n s o f p s e u d o - d i f f e r e n t i a l - o p e r a t o r t h e o r y . H e r e , w e g i v e a p r o o f w h i c h u s e s , i n s t e a d o f t h e m o l l i f i e r , t h e p o w e r o f L a p l a c i a n s a n d t h e n o t i o n o f i n d e x .

2 . L e t T b e a b o u n d e d o p e r a t o r o f a H i l b e r t s p a c e H 1 t o a H i l b e r t s p a c e H 2 . W e s a y t h a t T i s a n o p e r a t o r w i t h i n d e x i f t h e k e r n e l o f T a n d t h e c o k e r n e l o f T a r e f i n i t e - d i m e n s i o n a l . I f t h i s i s t h e c a s e , t h e i n d e x T i s d e f i n e d t ob e d i m k e r T — d i m c o k e r T . W e n e e d t h e f o l l o w i n g t w o p r o p o s i t i o n s :

P r o p o s i t i o n 1 . A s s u m e t h a t T i s a n o p e r a t o r w i t h i n d e x . L e t К b e a c o m p a c t o p e r a t o r o f H 1 to H 2 . T h e n T + К i s a n o p e r a t o r w i t h i n d e x ; a n d t h e i n d e x o f T i s e q u a l t o t h e i n d e x o f T + K .

P r o p o s i t i o n 2 . A s s u m e t h a t H 1 = H 2 a n d T i s H e r m i t i a n . T h e n , t h e i n d e x o f T i s z e r o , p r o v i d e d T i s a n o p e r a t o r w i t h i n d e x .

3 . W e l i s t h e r e p r o p e r t i e s o f S o b o l e v ' s n o r m s w h i c h w e n e e d . L e t F b e a v e c t o r s p a c e . I n p r a c t i c e , F w i l l b e t h e v e c t o r s p a c e o f f u n c t i o n s o f c l a s s С o v e r a c o m p a c t m a n i f o l d , o r , m o r e g e n e r a l l y , t h e v e c t o r s p a c e o f s e c t i o n s o f c l a s s С o f a v e c t o r b u n d l e o v e r a c o m p a c t m a n i f o l d . F o r e a c h s e R , t h e f i e l d o f r e a l n u m b e r s , w e a s s u m e t h a t a p r e - H i l b e r t s p a c e i n n e r p r o d u c t ( , ) s o n F i s g i v e n . W e s e t | ] u | | s = ( (u , u ) s) i W e d e n o t e b y ( F , s ) t h e p r e - H i l b e r t s p a c e F d e f i n e d b y t h e n o r m || ||s . D e n o t e b y H s ,o r H S ( F ) , t h e c o m p l e t i o n o f ( F , s ) . H s i s a H i l b e r t s p a c e , w h e r e t h e i n n e r p r o d u c t ( a n d t h e n o r m , r e s p e c t i v e l y ) i s a l s o d e n o t e d b y ( , )s ( r e s p . || ||s ) .

355

356 KURANISHI

A s s u m p t i o n (A ) : T h e i d e n t i t y m a p i : ( F , s ) -*■ ( F , t ) i s c o n t i n u o u s f o r s > t .

I f t h i s i s t h e c a s e , i i n d u c e s a u n i q u e c o n t i n u o u s m a p i : H s - * H t f o r s > t .

A s s u m p t i o n ( B ) : i : H s -* H t i s i n j e c t i v e f o r s > t .T h u s , w e m a y r e g a r d H s a s a v e c t o r s u b s p a c e o f H t a n d c o n s i d e r t h e

v e c t o r s p a c e H = U S H S.

A s s u m p t i o n (C ) : F = n s H s .L e t F 1 a n d F 2 b e v e c t o r s p a c e s o n w h i c h a r e g i v e n p r e - H i l b e r t s p a c e s

s a t i s f y i n g t h e a s s u m p t i o n s (A) to ( C ) . L e t

L : F 1 - F 2

b e a l i n e a r m a p . W e s a y t h a t L i s o f o r d e r m , w h e n

L : ( F 1 , s ) -* ( F 2 , s - m )

i s c o n t i n u o u s . I f t h i s i s t h e c a s e , L i n d u c e s a b o u n d e d l i n e a r m a p

L : H s ( F 1 ) -» H s _m ( F 2)

a n d h e n c e a l i n e a r m a p

L : H f F 1 ) - H ( F 2 )

2 3I f L ' : F -* F i s o f o r d e r m ' , i t i s c l e a r t h a t L ' o L i s o f o r d e r m + m ' .

T h e r e g u l a r i t y p r o p e r t y i n t h i s g e n e r a l c o n t e x t i s f o r m u l a t e d a s f o l l o w s : W e s a y t h a t L h a s t h e r e g u l a r i t y p r o p e r t y , w h e n L u e F a n d u E H f F 1 ) i m p l i e s t h a t u e F 1 . T o f o r m u l a t e o u r c o n d i t i o n s u n d e r w h i c h t h e r e g u l a r i t y p r o p e r t y h o l d s , w e h a v e t o i n t r o d u c e f u r t h e r a s s u m p t i o n s .

A s s u m p t i o n (D ) : T h e r e i s a p a i r i n g m a p

H © H 2 US H S © H . s -» Œ = t h e f i e l d o f c o m p l e x n u m b e r s

b y w h i c h H -s b e c o m e s t h e d u a l H i l b e r t s p a c e o f H s . T o b e m o r e p r e c i s e , d e n o t e b y t h e i m a g e o f u © w ( u se H s, w e H _ s) b y t h e a b o v e m a p . T h e n

<Ûj + u 2 , w)> = û-pW ^) + < ^ u 2 , w)> , ( « u , w У = a ^ u , w У



f o r a l l u 1( u 2 , u e H s , w e H . j j O ' e Œ . M o r e o v e r

I I S c s | |u | | , II w ||_s

w h e r e c s i s a c o n s t a n t , a n d f o r a n y v e H s t h e r e i s a u n i q u e w e H _ s s u c h t h a t <Cu,w)> = ( u , v ) s f o r a l l u e H s . W e a s s u m e f u r t h e r

 = (u , w ) Q

for a ll u, w e H0 .

IA EA -SM R -11/40 357

L e t F 1 a n d F 2 b e v e c t o r s p a c e s o n w h i c h a r e g i v e n p r e - H i l b e r t s p a c e s s a t i s f y i n g t h e a s s u m p t i o n s (A) to ( D ) . L e t

L : F 1 - F 2

b e a l i n e a r m a p o f o r d e r m . A l i n e a r m a p o f o r d e r m

L * . f 2 _ F i

i s c a l l e d t h e a d j o i n t o f L i f

< L u , w ^ = 

f o r a l l u E H j I F 1) a n d a l l w e H . s + m ( F 2 ). W e c a n s e e t h a t t h e a d j o i n t o f L e x i s t s u n i q u e l y a s f o l l o w s : P i c k w G H _ s+m( F 2 ). T h e n

H s( F 1) 3 u - * < L t t , w > e ( E

i s c l e a r l y a b o u n d e d m a p ( b e c a u s e | C L u , w)> | s с J| Lu. J|s_m l|w ||.s+m c ' | | u | | s | | w | | . s + m ), h e n c e b y a s s u m p t i o n (D) t h e r e i s a u n i q u e v e H ^ f F 1 ) s u c h t h a t

( L u , w)> = <(u, v)>

f o r a l l u E H s f F 1 ). W e s e t L * w = v a n d c h e c k L * i s t h e a d j o i n t o f L .D e n o t e b y L s t h e b o u n d e d m a p H ^ F 1 ) -* H s . m( F 2 ) i n d u c e d b y L . T h e n

t h e a d j o i n t o f L s ( a s a b o u n d e d m a p o f H J F 1 ) -*■ H s. m( F 2 )) e x i s t s , sa y ,( L s ) * . ( L s ) * i s a m a p o f H s_m ( F 2 ) -» H ^ F 1 ). W e n o t e h e r e t h a t ( L s ) * i sd i f f e r e n t f r o m L * . T h e f o r m e r i s b a s e d o n ( , ) s a n d ( , )s_m , b u t t h e l a t t e r i s b a s e d o n < , X

I f L 1: F 2 -*• F 3 i s o f o r d e r m 1, w e s e e e a s i l y t h a t

L * 0 L 1* = ( L 1 о L ) *

A s s u m p t i o n ( E ) : T h e r e i s a l i n e a r m a p

т ‘ : F -» F

o f o r d e r t s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n s :

i ) r 1 : H s+t ( F ) - H , ( F )

i s b i j e c t i v e a n d b i c o n t i n u o u s ,

i i ) r 1 : H t ( F ) - H 0 ( F )

i s a n i s o m o r p h i s m o f H i l b e r t s p a c e s ,

i i i ) t 1 о т г - r t+r ( a n d ( t 1) * - r 4, r e s p e c t i v e l y ) i s o f o r d e r t + r - 1 . ( r e s p . o f o r d e r t - 1 ) .

A s s u m p t i o n ( F ) : T h e i n j e c t i o n H S ( F ) -* H t ( F ) ( s > t ) i s a c o m p a c t l i n e a r m a p .

3 5 8 KURANISHI

B y a S o b e l e v s t r u c t u r e o f F w e m e a n a c o l l e c t i o n { ( , ) К .,У , t 51}s a t i s f y i n g a s s u m p t i o n s (A ) t o ( F ) .

L e t M b e a c o m p a c t m a n i f o l d a n d E b e a v e c t o r b u n d l e o v e r M . B y m e a n s o f a R i e m a n n i a n m e t r i c o n M , w e d e f i n e a p s e u d o - d i f f e r e n t i a l o p e r a t o r t $ : C°° ( M , E ) -* C “ ( M , E ) s u c h t h a t i t s p r i n c i p a l s y m b o l i s m u l t i p l i c a t i o n b y | | | s f o r a c o t a n g e n t v e c t o r Ç. T h e R i e m a n n i a n m e t r i c o n M t o g e t h e r w i t h a H e r m i t i a n m e t r i c o n E d e f i n e s a n i n n e r p r o d u c t ( , ) o n C “ ( M , E ) . D e f i n e ( , ) 0 b y К ..У . T h e n w e d e f i n e (, )s i n s u c h a w a y t h a t a s s u m p t i o n ( E ) i i ) h o l d s . T h e n w e s e e t h a t t h e s e f o r m a S o b o l e v s t r u c t u r e o n C ” ( M , E ) .

L e t F 1 , F 2 b e v e c t o r s p a c e s w i t h S o b o l e v s t r u c t u r e s . A l i n e a r m a p L : F 1 -» F 2 i s s a i d t o b e a h o m o m o r p h i s m ( o f o r d e r m ) o f S o b o l e v s t r u c t u r e s w h e n i t i s a m a p o f o r d e r m a n d t sL - L t s i s o f o r d e r m + s - 1 f o r a l l s i n IR. I f L : F 1 -» F 2 i s a h o m o m o r p h i s m o f o r d e r m , w e s e e e a s i l y b y a s s u m p t i o n ( E ) t h a t L * : F 2 -» F 1 i s a l s o a h o m o m o r p h i s m o f o r d e r m .I f L 1: F 2 ^ F 3 i s a h o m o m o r p h i s m o f o r d e r m ' , L 1 o L : F 1 F 3 i s a h o m o m o r p h i s m o f o r d e r m + m ' .

I n t h e c a s e o f S o b o l e v s t r u c t u r e s o n c ” ( M , E ) a n y p a r t i a l d i f f e r e n t i a l o p e r a t o r ( o r , m o r e g e n e r a l l y , p s e u d o - d i f f e r e n t i a l o p e r a t o r ) i s a h o m o m o r p h i s m o f S o b o l e v s t r u c t u r e s .

4 . O u r m a i n p u r p o s e i s to p r o v e t h e f o l l o w i n g r e g u l a r i t y t h e o r e m : L e t L b e a h o m o m o r p h i s m ( o f o r d e r m ) o f S o b o l e v s t r u c t u r e s o f F 1 t o F 2 . A s s u m e t h a t

( * ) Il L u U0 + IIuHh,.! ё с II u ||m ( u e F 1 )

( * * ) l | L ,;su ||0 + И v ||m_1 * с II v | | m ( v e F 2 )

f o r a c o n s t a n t с > 0 . T h e n L h a s t h e r e g u l a r i t y p r o p e r t y .

W e d i v i d e t h e p r o o f o f t h i s t h e o r e m i n s e v e r a l s t e p s :

1 2 i1) L e t L : F -*• F b e a h o m o m o r p h i s m o f o r d e r m . A s s u m e t h a t (w) h o l d s .T h e n f o r a n y s С IR

( - ) s II L u ||s + C s {J U ||s+m-l - C [J U ||s+m (u £ F )

f o r c o n s t a n t s C s > 0 . A p p l y i n g t h i s t o L * , w e s e e t h a t t h e a s s u m p t i o n ( * * ) i m p l i e s

( * * ), H L * v ||s + С,- H v ||i+m. 1 g c H v ||s+m (v e F 2 )

P r o o f . Il L u II s = | | t s L u ||0 ê II L t s u ||o - | | [ t s , L ] u ||0

ê CIIT4 L - lk su||m_1 - Il [TS, L ] U II 0

S i n c e t s ( a n d [ t s , L ] , r e s p e c t i v e l y ) i s o f o r d e r s ( a n d o r d e r s + m - 1,r e s p e c t i v e l y ) , i t f o l l o w s t h a t | | x s u Hm-i + || t T s , L ] a ||0 s C s || u Hs+nj-! .

2) D e n o t e b y k e r L n H s t h e k e r n e l o f L : H s i F 1 ) -* H s. m ( F 2 ). k e r L n H si s a c l o s e d H i l b e r t s u b s p a c e o f H ^ F 1) . k e r L П H s c l e a r l y i n c r e a s e s w i t h s .I f ( * ) h o l d s f o r L , t h e n k e r L П H s i s f i n i t e d i m e n s i o n a l .

IA EA-SM R-11/40 359

P r o o f , k e r L n H j b e i n g a c l o s e d s u b s p a c e o f H ^ F 1 ), w e r e g a r d i t a s a H i l b e r t s p a c e . B y a s s u m p t i o n ( F ) , t h e i n j e c t i o n k e r L n H s ->■ k e r L П H ^ i s a c o m p a c t l i n e a r m a p . O n t h e o t h e r h a n d , b y ( w h i c h i s v a l i df o r a l l v e H j f F 1 ) b y c o n t i n u i t y ) , w e s e e t h a t t h e n o r m || ||s a n d || o n k e r L n H s a r e e q u i v a l e n t . T h e n , b y a t h e o r e m i n B a n a c h s p a c e s t h e o r y , i t f o l l o w s t h a t t h e d i m e n s i o n o f k e r L n H s i s f i n i t e .

3) D e n o t e b y Y s t h e o r t h o g o n a l c o m p l e m e n t o f k e r L П H s . A s s u m e t h a t ( * ) h o l d s . T h e n f o r a c o n s t a n t c^ > 0

II L u | | s й с ' llull ( u £ Y ).l l ms s ii Ms + m ' s + m '

I n p a r t i c u l a r , t h e i m a g e o f L : H ^ ^ F 1 ) ->■ H S( F 2 ) i s a c l o s e d s u b s p a c e o f H S( F 2 ) . W e d e n o t e t h e i m a g e b y L H s+m.

P r o o f . I f t h e e s t i m a t e d o e s n o t h o l d , w e c a n f i n d a s e q u e n c e U y E H ^ ^ F 1 )

II Uv||s+m = 1 , ¡I L u v ||s 0

s u c h t h a t u v e Y s+m

S i n c e i : H ^ ^ F 1 ) -* H j ^ . ^ F 1) i s c o m p a c t , w e m a y a s s u m e t h a t u v i s a C a u c h y s e q u e n c e i n H ^ ^ ^ F 2 ) b y r e p l a c i n g i t b y a s u b s e q u e n c e i f n e c e s s a r y . S i n c e ( * ) s h o l d s f o r u e H ^ I F 1) b y c o n t i n u i t y , i t f o l l o w s t h a t u v i s a C a u c h y s e q u e n c e i n H s+m( F 1) . T h u s u v c o n v e r g e s t o a n e l e m e n t u i n H ^ ^ F 1 ).

L u = 0

T h i s i s a c o n t r a d i c t i o n ( b e c a u s e Y s+m n k e r L = { 0 } ) . T h e r e f o r e c ’s a s a b o v e e x i s t s . O n c e t h e e s t i m a t e i s e s t a b l i s h e d , i t i s e a s y t o s e e t h a t L H s+m i s a c l o s e d s u b s p a c e o f H S( F 2).

4 ) S e tX s = { v e H s ( F 2) ; ( v , z ) = 0 f o r a l l z £ k e r L * n H_s }

A s s u m e t h a t ( * * ) h o l d s . T h e n

P r o o f . D e n o t e b y Y ' t h e o r t h o g o n a l c o m p l e m e n t o f k e r L * n H s i n H S( F 2 ). B y 3 ) a p p l i e d t o L * , w e s e e t h a t L * H _ s+m = L * Y l s+m i s a c l o s e d s u b s p a c e o f H . J F 1 ) . P i c k v e X s . m . S i n c e f o r a n y u e L * H s+m t h e r e i s a u n i q u ey e Y . s+m w i t h u = L * y , w e c a n d e f i n e t h e l i n e a r m a p

4>- L>:' H -s+m Э u = L * y « < y , v > e <C

w h e r e y e Y^s+m . B y t h e e s t i m a t e i n 3) a p p l i e d t o L * w e f in d t h a t = ( L * y , u ' ) = < T _sL * y , t ’ s u ' >

= < y , L ( t ’ s ) * t s u ' > = < y , L u >

360 KURANISHl

w h e r e u = ( t ' s ) * t " s u ' e H . s+2s ( F 1) = H j i F 1 ). T h u s < y , v - L u ) = 0 f o r a l l

y e Y -s+m- S i n c e v e x s-m a n d L H s Q X s-m ' < z . v - L u > = 0 f o r a l l z e k e r L * n H . s+m- S i n c e H . s+m( F 2 ) = ( k e r L * n H . s+m) + Y ! s+nv i t f o l l o w s t h e n t h a t < w , v - L u ) = 0 f o r a l l w e H _ s+m( F 2), i . e . v = L u .

5) A s s u m e t h a t F 1 = F 2 = F , L s a t i s f i e s ( * ) , a n d t h a t L * = L + R , w h e r e R i s o f o r d e r m - 1 . T h e n

d i m ( H s_m / L H s ) = d i m ( k e r L n H s )

P r o o f . C o n s i d e r t h e b o u n d e d l i n e a r m a p

L : H s - H s . m

w h i c h w e d e n o t e b y L s . B y o u r a s s u m p t i o n L s a t i s f i e s ( * ) a n d ( * * ) . H e n c e b y 2) a n d 4 ) , L s i s w i t h i n d e x . W e h a v e to s h o w t h a t t h e i n d e x o f L s = 0 .S i n c e r ‘ m : H s_m -> H j i s a h o m e o m o r p h i s m o f t o p o l o g i c a l v e c t o r s p a c e , th e i n d e x o f L s i s e q u a l t o t h a t o f

t - ™ L s : H s - H s

W e d e n o t e b y ( r ‘ m L ) * t h e H i l b e r t s p a c e a d j o i n t o f t h e a b o v e m a p . In v i e w o f p r o p o s i t i o n s i n 2 . , o u r c o n t e n t i o n i s p r o v e d i f w e s h o w t h a t

( r ' m L s ) * = r " m L s + К

w h e n К i s a c o m p a c t o p e r a t o r . F o r u, v G H s

(u, (T‘ mL s ) * v ) s = ( r 'mL u , v)j = <т“ r " mL u , rs v >

= <tsu, ((r~)~'Y'l. : (t":!: )” [r* ) v>

= (u, ( r 5)’ 1 ((T ') '1 ) * L * ( T - ,n) * (T , ) V v )1

H e n c e

(T-m L , ) * v = ( t T 1 ( ( t s ) " 1 ) " I / <( t ‘ !:; ) ' ( t s r V - v

O n t h e o t h e r h a n d , a s s u m p t i o n ( E ) a n d o u r a s s u m p t i o n i n 5) i m p l y t h a t t h e r i g h t - h a n d s i d e o f t h e a b o v e i s e q u a l t o T ‘ m L + К w h e r e К i s o f o r d e r - 1 , q . e . d .

6 ) U n d e r t h e a s s u m p t i o n s i n 5 ) , k e r L n H s a n d k e r L * n H s a r e i n d e p e n d e n t o f s . I n p a r t i c u l a r , k e r L n H , a n d k e r L * n H s a r e i n F .

P r o o f . B y 4 ) d i m H s_m / L H s = d i m ( k e r L * n H . s+m). H e n c e d i m (H s. m / L H s ) i n c r e a s e s w i t h s . O n t h e o t h e r h a n d , d i m ( k e r L n H s ) d e c r e a s e s a s s i n c r e a s e s . T h e n w e f in d b y 5) t h a t d i m ( k e r L n H s ) i s i n d e p e n d e n t o f s .S i n c e k e r L n H s Q k e r L n H t f o r s > t , i t f o l l o w s t h e n k e r L n H s i s i n d e p e n d e n t o f s . T h e n k e r L n H s c F b y a s s u m p t i o n ( C ) . S i n c e L * a l s o s a t i s f i e s t h e c o n d i t i o n i n 5 ) , w e s e e k e r L * л Hs i s i n d e p e n d e n t o f s a n d k e r L * л H s c F .

7) U n d e r t h e a s s u m p t i o n i n 5) t h e r e g u l a r i t y p r o p e r t y f o r L h o l d s .

IA EA-SM R-11/40 361

P r o o f . B y 4 ) , L H S = X s_m . O n t h e o t h e r h a n d , k e r L * n H s = k e r L * n Г b y 6 ) . T h u s

L H S = ( т е H s_m ; < v , z > = 0 f o r a l l z e k e r L * n F }

B y l o o k i n g a t t h e r i g h t - h a n d s i d e o f t h e a b o v e , w e c o n c l u d e t h a t

L H t n H s. m = L H S

f o r a l l t s s . T h e n , i n v i e w o f 6 ) w e e a s i l y s e e t h e f o l l o w i n g : L u £ H s . m i m p l i e s t h a t u e H s . T h e r e g u l a r i t y p r o p e r t y i s a c o n s e q u e n c e o f t h i s c o n c l u s i o n .

8) L e t L b e a h o m o m o r p h i s m ( o f o r d e r m ) o f S o b o l e v s t r u c t u r e s o f F 1 t o F 2. A s s u m e t h a t L s a t i s f i e s t h e c o n d i t i o n s ( * ) a n d ( * * ) . T h e n t h e r e g u l a r i t y p r o p e r t y h o l d s f o r L .

P r o o f . L u = v i m p l i e s L * L u = L * v . H e n c e t h e r e g u l a r i t y p r o p e r t y o f L f o l l o w s f r o m t h a t o f L * L . O n t h e o t h e r h a n d , w e e a s i l y s e e t h a t t h e a s s u m p t i o n i n 5 ) f o r L * L ( t h u s , m t h e r e s h o u l d b e r e p l a c e d b y 2 m ) f o l l o w s f r o m t h e c o n d i t i o n ( * ) a n d ( * * ) f o r L .

T h e e l l i p t i c p a r t i a l d i f f e r e n t i a l o p e r a t o r s o f o r d e r m a r e k n o w n to s a t i s f y c o n d i t i o n s ( * ) a n d ( * * ) . H o w e v e r , S o b o l e v n o r m s s a t i s f y c o n d i t i o n s (A) t o ( F ) o n l y w h e n w e c o n s i d e r f u n c t i o n s o v e r c o m p a c t m a n i f o l d s . T h e r e f o r e , b y t h e a b o v e c o n c l u s i o n , w e s e e t h a t e l l i p t i c d i f f e r e n t i a l o p e r a t o r s o n c o m p a c t m a n i f o l d s h a v e t h e r e g u l a r i t y p r o p e r t y . H o w e v e r , w e c a n o b t a i n t h e l o c a l r e g u l a r i t y t h e o r e m o u t o f t h e g l o b a l r e g u l a r i t y t h e o r e m . T o m a k e t h e m a t t e r e l e m e n t a r y , l e t u s s a y t h a t , f o r a d o m a i n П i n R n, a m e a s u r a b l e f u n c t i o n u b e l o n g s t o H s (f2) i f a n d o n l y i f f u e H s ( R n ) f o r a n y C ° ° - f u n c t i o n f w i t h s u p p o r t in a b o u n d e d d o m a i n w h e r e i s d i f f e o m o r p h i c toa b a l l . L e t L b e a n e l l i p t i c p a r t i a l d i f f e r e n t i a l o p e r a t o r o f o r d e r m o n Í2. A s s u m e t h a t u e H s ( Q) , s a y s Ш m , a n d L u = v i s o f c l a s s C ° ° . N o w , w e a s s e r t t h a t u i s o f c l a s s C " . N a m e l y , c h o o s e f a s a b o v e , a n d e m b e d f ix in a c o m p a c t m a n i f o l d M o f d i m e n s i o n n , s a y , a t o r u s . C o n s t r u c t a n e l l i p t i c p a r t i a l d i f f e r e n t i a l o p e r a t o r , s a y L 1; o n M w h i c h c o i n c i d e s w i t h L f o r p o i n t s in K , w h e r e К i s t h e s u p p o r t o f f . S i n c e

L f u = f v + [ L , f ] u

a n d [ L , f ] i s o f o r d e r m - 1, i t f o l l o w s t h a t L j^ ( fu ) e H s . m+1 (M ) . H e n c e , b y t h e g l o b a l v e r s i o n , fu e H S+1( M ) . T h e r e f o r e , u e H s+1(£2). T h e n , b y i n d u c t i o n , w e c a n c o n c l u d e t h a t u i s o f c l a s s C “ .

SECRETARIAT OF SEMINAR

ORGANIZING COM M ITTEE

M . D o l c h e r

J . E e l l s

J . C . Z e e m a n

I s t i t u t o d i M a t e m a t i c a , U n i v e r s i t á d i T r i e s t e , I t a l y

I n s t i t u t e o f A d v a n c e d S t u d i e s ,P r i n c e t o n , N . J . , U n i t e d S t a t e s o f A m e r i c a

I n s t i t u t e o f M a t h e m a t i c s , U n i v e r s i t y o f W a r w i c k , C o v e n t r y , W a r k s , U n i t e d K i n g d o m

E D I T O R I A L B O A R D O F P R O C E E D I N G S

P . d e l a H a r p e

M . D o l c h e r

J . E e l l s

I n s t i t u t e o f M a t h e m a t i c s , U n i v e r s i t y o f W a r w i c k , C o v e n t r y , W a r k s , U n i t e d K i n g d o m

I s t i t u t o d i M a t e m a t i c a , U n i v e r s i t é d i T r i e s t e , I t a l y

I n s t i t u t e o f A d v a n c e d S t u d i e s ,P r i n c e t o n , N . J . , U n i t e d S t a t e s o f A m e r i c a

J . C . Z e e m a n I n s t i t u t e o f M a t h e m a t i c s , U n i v e r s i t y o f W a r w i c k , C o v e n t r y , W a r k s , U n i t e d K i n g d o m

E D I T O R S

A . M . H a m e n d e

J . W . W e i l

I n t e r n a t i o n a l C e n t r e f o r T h e o r e t i c a l P h y s i c s , T r i e s t e , I t a l y

D i v i s i o n o f P u b l i c a t i o n s , I A E A ,V i e n n a , A u s t r i a

363

HOW TO ORDER IAEA PUBLICATIONS

Exclusive sales agents for IAEA publications, to whom all ordersand inquiries should be addressed, have been appointed in the following countries:

U N I T E D K I N G D O M H er Majesty's Stationery Office, P .O . Box 569, Lo n d o n S E 1 9 N H

U N I T E D S T A T E S O F A M E R I C A U N I P U B , Inc., P.O. Box 433, M u rra y Hill Station, N e w Y o r k , N . Y . 10016

■ In the following countries IAEA publications may be purchased from the sales agents or booksellers listed or through your major local booksellers. Payment can be made in local currency or with UNESCO coupons.

A R G E N T I N A Comisión Nacional de Energía A tóm ic a, Avenida del Libertador 8250, Buenos Aires

A U S T R A L I A H unter Publications, 58 A Gipps Street, Co l li n gw oo d, Victoria 3 066 B E L G I U M Office International de Librairie, 30, avenu eM arnix, B -1 0 5 0 Brussels C A N A D A Inform ation Canada, 171 Slater Street, O ttaw a, O n t . К 1 A O S 9

C.S.S.R. S . N . T . L . , Spálená 51, C S - 110 00 PragueAlfa, Publishers, H urb an ov o námestie 6, C S -8 0 0 0 0 Bratislava

F R A N C E Off ice International de D ocum entation et Librairie, 48, rue Gay-Lussac, F -7 5 0 0 5 Paris

H U N G A R Y Kultura, Hungarian T rad ing C o m p a n y for B ook s and Newspapers,P.O. B ox 149, H -101 1 Budapest 62

I N D I A O x f o r d B ook and Stationery C o m p . , 17, Park Street, Calcutta 16 I S R A E L Heiliger and Co. , 3, Nathan Strauss Str., Jerusalem

I T A L Y Librería Scientifica, Dott . de Biasio Lu cio "a e io u ” ,Via Meravigli 16, 1-20123 Milan

J A P A N Maruzen C o m p a n y , Ltd ., P .O .B o x 5050, 100-31 T o k y o International N E T H E R L A N D S Marinus N ijhoff N . V . , Lange V o o r h o u t 9-11 , P .O . Box 269, T h e Hague

P A K I S T A N Mirza B ook Ag ency , 65, T h e Mall, P .O .B o x 729, Lahore-3 P O L A N D Ars Polona, Céntrala Han dlu Zagranicznego, Krakowskie Przedmiescie 7,

WarsawR O M A N I A Cartimex, 3-5 13 Dec em brieStreet, P .O .B o x 134-13 5, Bucarest

S O U T H A F R I C A Van Schaik's Bookstore, P .O .B o x 724, PretoriaUniversitas B ook s (P ty) Ltd., P .O .B o x 1557, Pretoria

S P A IN Nautrónica, S .A . , Pérez A y u s o 16, Madrid -2 S W E D E N C .E . Fritzes Kungi . Hovbokhandel, Fredsgatan 2, S -1 03 07 S toc k h olm

U.S .S .R . Mezhdunarodnaya Kniga, Smol en skaya-Sennaya 32-34, M o scow G -2 0 0 Y U G O S L A V I A Jugoslovenska Knjiga, Terazije 27, Y U - 1 1000 Belgrade

Orders from countries where sales agents have not yet been appointed and requests for information should be addressed directly to:

Publishing Section,International Atomic Energy Agency,Kàrntner Ring 11, P.O.Box 5 90 , A-1011 Vienna, Austria

IN TER N ATIO N A L A TO M IC ENERGY A G E N C Y V IE N N A , 1974

PRICE: US $ 17 .00A u strian S ch illin gs 3 2 0 ,- (£7.20; F .Fr. 8 2 ,- ; DM 4 5 ,- )

SUBJECT GROUP: III Physics /Theoretica l Physics

Date post:	31-Jan-2023
Category:	Documents
Upload:	khangminh22
View:	0 times
Download:	0 times

INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1974

Documents