Zheng-Xu He and Oded Schramm- The C^infinity -convergence of hexagonal disk packings to the Riemann...

8/3/2019 Zheng-Xu He and Oded Schramm- The C^infinity -convergence of hexagonal disk packings to the Riemann map

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Ac ta Math. , 180 (1998), 219-245(~) 1998 by Institut Mittag-Leffier. All rights reserved

Th e C -convergence o f hex agon ald i sk pack ings t o t he R iemann map

Z H E N G - X U H EUniversity of California, San Diego

La Jolla, CA, U.S.A.

bya n d O D E D S C H R A M M

The Weizmann Institute of ScienceRehovot, Israel

1 . I n t r o d u c t i o nL e t ~t~ CC b e a s i m p l y - c o n n e c t e d d o m a i n . T h e R o d i n - S u l l i v a n T h e o r e m s t a t e s t h a t as e q u e n c e o f d i s k p a c k i n g s i n t h e u n i t d i s k U c o n v e r g e s , i n a w e l l - d e f in e d s e n s e , t o ac o n f o r m a l m a p f r o m f ~ t o U . M o r e o v e r , i t is k n o w n t h a t t h e f i rs t a n d s e c o n d d e r i v a t i v e sc o n v e r g e a s w e ll . H e r e , i t is p r o v e n t h a t f o r h e x a g o n a l d i s k p a c k i n g s t h e c o n v e r g e n c ei s C~ T h i s i s d o n e b y s t u d y i n g M h b i u s i n v a r i a n t s o f t h e d i s k p a c k i n g s t h a t a r e d i s c r e t ea n a l o g s o f th e S c h w a r z i a n d e r i v a t iv e .

A s a c o n s eq u e n c e , t h e f i r st n d e r i v a t iv e s o f t h e c o n f o r m a l m a p c a n b e a p p r o x i m a t e db y q u a n t i t i e s w h i c h d e p e n d o n t h e p o s i t i o n s o f t h e c e n t e r s o f s o m e n 1 c o n s e c u t i v e d i sk si n t h e p a c k i n g .

S u p p o s e t h a t P = ( P v : v E V ) i s a n i n d e x e d d i s k p a c k i n g i n t h e p l a n e . T h e nerve ( o rtangency graph) o f P i s t h e g r a p h G = ( E , V ) o n V , t h e i n d e x i n g s e t o f P , s u c h t h a t[ v , u ] C E i f a n d o n l y i f P v a n d P u a r e t a n g e n t . T h e D i s k P a c k i n g T h e o r e m [ 1 0] s a y st h a t g i v e n a g r a p h G w h i c h i s t h e 1 - s k e le t o n o f a t o p o l o g i c a l t r i a n g u l a t i o n o f U = { z C C :I z l ~ < l } , t h e r e i s s o m e d i s k p a c k i n g i n U w i t h n e r v e G , s u c h t h a t t h e b o u n d a r y d i s k s( i. e. , t h o s e d i s k s c o r r e s p o n d i n g t o t h e b o u n d a r y v e r t i c e s ) a r e a ll t a n g e n t t o t h e u n i tc i r c l e OU. M o r e o v e r , t h e d i s k p a c k i n g i s u n i q u e u p t o ( p o s s i b l y o r i e n t a t i o n - r e v e r s i n g )M h b i u s t r a n s f o r m a t i o n s o f U ( s e e , f o r e x a m p l e , [ 18 , C h a p t e r 1 3] , [1 2] , [3] o r [ 4] f o r o t h e rp r o o f s a n d e x t e n s i o n s ) .

F o r e a c h s > 0 , l e t H ~ d e n o t e t h e h e x a g o n a l g r i d w i t h m e s h ~ . T h e v e r t i c e s o f H ~f o r m t h e h e x a g o n a l l a t t i c e :

V E= { n e + m w ~ : ( n , m ) 9 Z Z} ,The f ir s t named author w as suppor ted by NSF Grant DMS 96-22068, and the second named authorby NSF Grant DMS 94-03548.


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220 Z.-X. HE AND O. SC HRAMM

Fig. 1.1. Th e packings R e and Pe, and the map fe. Several points have been marked to aidin grasping the correspondence.

w h e r e w is t h e c u b e r o o t o f - 1 ,w---- ex p 89 = 89 x / 3 i

a n d a n e d g e c o n n e c t s a n y t w o v e r t i ce s o f H E a t d i s t a n c e rL e t f~ b e a s i m p l y - c o n n e c t e d d o m a i n i n C w i t h 1 2 ~ C . T h e n t h e r e i s a s u b g r id H ~

o f H e , e q u a l t o t h e 1 - s k e l e to n o f a t r i a n g u l a t i o n o f a c l o s e d t o p o l o g i c a l d i s k c o n t a i n e d i nt h a t a p p r o x i m a t e s ~ . L e t V ~ d e n o t e th e s e t o f v e r ti c e s i n H ~ . F r o m t h e D i sk P a c k i n g

E. ET h e o r e m i t f o ll o w s t h a t t h e r e is a d is k p a c k i n g P = ( P ~ . v E V ~ ) i n 0 w h o s e n e r v e isH ~ a n d w i t h t h e p r o p e r t y t h a t b o u n d a r y d is k s a r e t a n g e n t t o O U . F o r e a c h v E V ~ ,le t r E ( v ) d e n o t e t h e c e n t e r o f t h e d i s k P ~ ( s e e F i g u r e 1 . 1) . T h e n t h e R o d i n - S u l l i v a nT h e o r e m [ 15 ] t e l ls u s t h a t , a s s u m i n g t h a t t h e p a c k i n g s p E a r e s u i t a b l y n o r m a l i z e d ( b y aM 6 b i u s t r a n s f o r m a t i o n , p o s s i b l y o r i e n t a t i o n - r e v e r s i n g ) , t h e d i s c r e t e f u n c t i o n s fE : V ~ - * Uc o n v e r g e " l o c a l l y u n i f o r m l y " i n f~ t o t h e ( s i m i l a r l y n o r m a l i z e d ) R i e m a n n m a p f : ~ - - * U ,w h e n e - * 0 .

T h e r e i s a n a t u r a l d e f i n i t io n f o r t h e d i s c r e t e p a r t i a l d e r i v a t i v e s o f f u n c t i o n s d e f i n e do n t h e l a t t i c e V ~. G i v e n d i s c r e t e f u n c t i o n s g E : V ~ - - * C , w h e r e e 6 ( 0 , 1 ), w e s a y t h a tgE converge in C~176 t o a s m o o t h f u n c t i o n g : f ~ - * C a s e - - * 0 , i f g E c o n v e r g e l o c a l l yu n i f o r m l y i n f~ t o g , a n d t h e d i s c r e t e p a r t i a l d e r i v a t i v e s o f gE o f a n y o r d e r c o n v e r g el o c a l l y u n i f o r m l y t o t h e c o r r e s p o n d i n g p a r t i a l d e r i v a t i v e s o f g . T h e p r e c i s e d e f i n i t io n sw i l l be g i ven i n w

F o r e a c h v 6 V ~ , l e t rE(v) d e n o t e t h e r a d i u s o f P ~ . O u r m a i n t h e o r e m is :C ~176 TH EO R EM 1 .1 . T h e d i s c re t e f u n c t i o n s d E: V ~ - ~ U c o nv e rg e i n

C ~ ( g t ) t o t he R i em an n m a pp i n g f : 12---~U . The d i s c re t e f un c t i o ns 2rE ( v ) / r converge i nC ~ ( a ) to I f ' l -


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C C ~ - C O N V E R G E N C E O F D I S K P A C K I N G S 22 1

A s a c o n s e q u e n c e , t h e d e r i v a t i v e s o f t h e c o n f o r m a l m a p u p t o o r d e r n c a n b e a p -p r o x i m a t e d b y q u a n t i t ie s w h i c h d e p e n d o n t h e p o s i t i on s o f th e c e n t e rs o f s o m e n c o n s e c u t i v e d i s k s i n t h e p a c k i n g .

W . T h u r s t o n [ 19 ] c o n s t r u c t e d t h e p a c k i n g s p E , a n d c o n j e c t u r e d t h a t f ~ c o n v e r g e st o t h e R i e m a n n m a p p i n g . T h e l o c a l l y u n i f o r m c o n v e r g e n c e ( i. e ., C ~ w a sp r o v e d b y R o d i n a n d S u l l iv a n [ 1 5] , u s i n g a r i g i d i t y t h e o r e m f o r q u a s i - c o n f o r m a l d e f o r m a -t i o n s o f S c h o t t k y g r o up s a n d a l e n g t h - a re a a r g u m e n t . T h e C l - c o n v e r g e n c e w a s p ro v e di n [ 6 ], u s i n g a n a r e a e s t i m a t e ; a n d C 2 - c o n v e r g e n c e i n [ 5 ], u s i n g t h e s a m e a r e a e s t i m a t e .T h e r e s u l t o f [5] s a y s t h a t t h e " i n t e r s t i c ia l m a p s " c o n v e r g e in C 2 t o f . I n [8] i t i s s h o w nt h a t t h e m e t h o d o f [ 6 ] w o r k s w e l l f o r g e n e r a l d i s k p a c k i n g s w i t h b o u n d e d v a l e n c e .

R e c e n t l y , t h e C 2 - c o n v e r g e n c e o f t h e i n t e r s t i c i a l m a p s h a s b e e n g e n e r a l i z e d i n [9]t o d i s k p a c ki n g s o f a r b i t r a r y c o m b i n a t o ri a l p a t t e r n , e v e n w i t h o u t t h e a s s u m p t i o n o fb o u n d e d v a l e n c e . T h i s i s d o n e u s i n g m e t h o d s o f d i s c r e t e e x t r e m a l l e n g t h a n d f i x e d p o i n ti n d e x a r g u m e n t s . W e i n t e n d t o w r i t e a p a p e r w h i c h f u r t h e r s i m p l i fi e s [9 ] , a v o i d s t h ed i s c r e t e e x t r e m a l l e n g t h m e t h o d s , a n d g i v e s e s t i m a t e s f o r t h e c o n v e r g e n c e r a t e s .

S ev era l o th e r i n t e r es t i n g r esu l t s a l so ap p eare d i n [1 3 ] , [ 1 4 ], [1] an d [2] . Th e se w o rk sm a y b e c o m b i n e d t o g e t h e r t o y i e ld a n a l t e r n a t i v e p r o o f o f t h e C l - c o n v er g e n c e . W e n o t et h a t K . S t e p h e n s o n [ 17 ] h a s a l s o g i v e n a p r o o f o f t h e C ~ u s i n g M a r k o v c h a i n sa n d e l e c t r ic a l n e t w o r k s " w i t h l e a k s " ; a n d r e c e n t l y , L . C a r l e s o n h a s f o u n d a d i f f e r e n t p r o o fb a s e d o n R o d i n ' s e q u a t i o n [ 1 3] , n a m e l y ,

6

k = l R ( r k + r k + l + R ) '

w h e r e r k , k E Z 6 = Z / 6 Z , a r e t h e r a d i i o f s ix d is k s w h ic h s u r r o u n d a d i s k o f r a d i u s R .F o l l ow i n g i s a b r i e f d e s c r i p t i o n o f o u r m e t h o d . F i r s t s o m e M h b i u s i n v a r i a n t s f o r

f i n it e h e x a g o n a l d i s k p a c k i n g s a r e d e f in e d a n d t h e i r e l e m e n t a r y e q u a t i o n s a r e d e r i v e d .S i m i l a r i n v a r i a n t s a n d e q u a t i o n s w e r e i n t r o d u c e d [ 16 ] i n t h e s e t t i n g o f c ir c le p a t t e r n sw i t h t h e c o m b i n a t o r i c s o f t h e s q u a r e g r i d . W e w i l l t h e n u s e t h e M h b i u s i n v a r i a n t s t od e f in e ( d i s c r e te ) S c h w a r z i a n s ( o r S c h w a r z i a n d e r i v a ti v e s ) . R o u g h l y , t h e S c h w a r z i a n s a r es o m e a p p r o p r i a t e l y s c a l e d m e a s u r e o f t h e d e f o r m a t i o n o f p a i r s o f n e i g h b o r i n g i n t e r s t ic e sf r o m t h e i r s t a n d a r d p o s i ti o n s a n d , a s t h e c o n t i n u o u s S c h w a r z ia n , t h e y a r e u n c h a n g e d i ft h e p a c k i n g i s r e p l a c e d b y a M h b i u s i m a g e . F o r a r e g u l a r h e x a g o n a l p a c k i n g ( i n w h i c ha l l d i s k s h a v e id e n t i c a l s i ze ) , t h e S c h w a r z i a n s a r e t a i l o r e d t o b e 0 . T h e r e s u l t s o f [6]w i l l b e u s e d t o s h o w t h a t t h e S c h w a r z i a n s o f t h e p a c k i n g s P ~ a r e b o u n d e d . O n t h eo t h e r h a n d , f r o m t h e e q u a t i o n s s a t i s f i e d b y t h e M h b i u s i n v a r i a n t s , w e w i l l s h o w t h a t t h e( d i s c re t e ) L a p l a c i a n o f th e S c h w a r z i a n s i s a p o l y n o m i a l f u n c t i o n in ~ a n d t h e S c h w a r z i a n st h e m s e l v e s . I t i s t h e n a c o n s e q u e n c e o f a r e g u l a r i t y t h e o r e m o f d i s c r e t e e l li p t ic e q u a t i o n s


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222 Z.-X. HE AND O. SCHRAMM

t h a t a l l d i s c r e t e d e r i v a t i v e s o f t h e S c h w a r z i a n s , o f a n y g i v e n o r d e r , a r e l o c a l l y b o u n d e d .T h e C ~ - c o n v e r g e n c e o f f~ a n d 2 r ~ / ~ w i l l t hen f o l l ow .

I t is a ls o p r o v e d t h a t a l i n e a r c o m b i n a t i o n o f t h e d i s c r e t e S c h w a r z i a n s o f P e c o n v e r g e si n C ~r t o t h e S c h w a r z i a n d e r i v a t i v e o f t h e c o n f o r m a l m a p p i n g . A n o t h e r i n t e r e s t i n gr e s u l t is t h a t s o m e n a t u r a l " c o n t a c t t r a n s f o r m a t i o n s " d e f i n e d b y t h e p o s i t i o n s o f t r i p l e t so f m u t u a l l y t a n g e n t d i s k s c o n v e r g e i n C ~ I n f a c t, a n y r e a s o n a b l e , n a t u r a l d i s c r e t ef u n c t i o n a s s o c ia t e d t o P ~ c a n b e s h o w n t o b e c o n v e r g e n t in C ~

I t i s p o s s i b l e t o c a r r y o u t o u r p r o o f w i t h a n i n v e s t i g a t i o n o f t h e r a d i i i n p la c e o f t h eS c h w a r z i a n s , a n d w i t h R o d i n ' s e q u a t i o n ( 1 .1 ) ta k i n g t h e p l a c e o f t h e e q u a t i o n w h i c h t h ed i s c r e te S c h w a r z i a n s sa t is f y. B u t p e r h a p s s o m e o f th e d e t a il s b e c o m e m o r e c o m p l i c a te d .

T h e r e s t o f t h e p a p e r i s o r g a n i z e d a s f o ll o w s. w i n t r o d u c e s s o m e d i s c r e t e p a r t i a ld i f f e r e n ti a l o p e r a t o r s o n t h e l a t t i c e s V ~, a n d d e f i n e s p r e c i s e l y t h e m e a n i n g o f lo c a l l yu n i f o r m c o n v e r g e n c e a n d C ~ w g iv e s t h e c o n s t r u c t i o n o f t h e s u b g r i d H ~a n d f i x e s a n o r m a l i z a t i o n f o r P ~ , a n d h e n c e f o r f ~ a n d f . w i n t r o d u c e s t h e S c h w a r z i a n s ,a n d d e r i v e s t h e e q u a t i o n s s a t i sf i e d b y t h e m . T h e s e e q u a t i o n s w i l l b e u s e d i n w t o o b t a i nt h e f o r m u l a f o r t h e d i s c r e t e L a p l a c i a n o f t h e S c h w a r z i a n s . A c o n s e q u e n c e o f t h e f o r m u l ai s t h a t i f t h e S c h w a r z i a n s a r e a l l b o u n d e d , t h e n s o a r e t h e i r L a p l a c i a n s . I n w w e w i llp r o v e t h a t t h e S c h w a r z ia n s a r e u n i f o r m l y b o u n d e d i n a n y c o m p a c t s u b s e t o f f~ . I n wt h e L i p s c h i t z n o r m o f a d i s c r e t e d e r i v a t i v e o f a f u n c t i o n i s e s t i m a t e d i n t e r m s o f t h e L ~n o r m s o f t h e f u n c t io n a n d i ts d i s c r e te L a p l a c ia n . A t t h a t p o i n t , o n e c a n c o n c l u d e t h a tt h e S c h w a r z i a n s a r e u n i f o r m l y L i p s c h i t z , a n d t h e r e f o r e c o n v e r g e n t i n C ~ I n w l o c a lu n i f o r m b o u n d s o n t h e p a r t i a l d e r i v a ti v e s o f a n y o r d e r f o r t h e S c h w a r z i an s a r e o b t a i n e d .I n w t h e c o n t a c t t r a n s f o r m a t i o n s o f d i s k p a t t e r n s a r e d e f i n e d , a n d i t i s s h o w n t h a t t h e yc o n v e r g e i n C ~ T h e n , i n w t h e m a i n t h e o r e m is o b t a i n e d a s a c o n s e q u e n c e o f t h eC ~ - c o n v e r g e n c e o f t h e c o n t a c t t r a n s f o r m a t i o n s . F i n a ll y , d is k p a t t e r n s o f m o r e g e n e r a lc o m b i n a t o r i c s a r e d i s c u s s e d i n w

W e t h a n k t h e a n o n y m o u s r e f er e e f o r v a l u a b le s u g g e s ti o n s, w h i c h l e a d u s t o t h ec u r r e n t f o r m u l a t i o n o f t h e m a i n t h e o r e m .

2 . P r e l i m i n a r i e s o n d i s c r e t e d i ff e r e n t i a l o p e r a t o r sa n d t h e c o n t i n u o u s l i m i t o f d i s c r e t e f u n c t i o n s

W e w i l l n e e d t o i n t r o d u c e s o m e d i s c r e t e d i f f e r e n ti a l o p e r a t o r s . F o r e a c h e > 0 , r e c a l l t h a tt h e s e t V ~ i s t h e s e t o f v e r t i c e s in t h e h e x a g o n a l g r i d H e , i .e . , th e p o i n t s n e + m w e , w h e r ew = e xP( 89 and n , m a r e i n t e g e r s .

F o r a n y k E Z 6 , l e t L ~ : V e - - ~ V ~ d e n o t e t h e t r a n s l a t i o nL ~k v = v + ~ w k . ( 2 .1 )


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C ~ - C O N V E R G E N C E O F D IS K P A C K I N G S 223

L e t W C _ V ~ b e a s u b s e t . A v e r t e x v C W i s c a l l e d a n i n t e r i o r v e r t e x o f W i f f o re a c h k , k E Z 6 , t h e n e i g h b o r i n g v e r t e x L ~ v i s c o n t a i n e d i n W . L e t W ( ~ a n d f o r e a c hi n t e g e r / ~ > 1 , l e t W ( 0 d e n o t e t h e s e t o f i n t e r i o r v e r t i c e s o f W ( 1 -1 ) .

G i v e n a f u n c t i o n ~?: W - * R , t h e ( d i s c r e t e ) d i re c t i o n a l d e r i v a t i v e 0 ~ : W O ) - ~ R i sd e f i n e d b y

c3~w(v) = ~ -1 (7](L ~ v ) - v ( v ) ) = ~ - l ( ? 7 ( v _ ~ _ c ~ d k ) _ T ] ( v ) ) . (2 .2)L e t L~cW d e n o t e t h e f u n c t i o n w h i c h d i f f er s f r o m 7? b y t h e t r a n s l a t i o n L ~r

L ~ ( v ) = ~ ( L ~ v ) = y ( v + ~ w k ) . (2 .3)

T h e n w e h a v e 0~r = ~- 1 [L~ - I] ,w h e r e I ~ - - ~ ? .

T h e ( d i s c r e t e ) L a p l a c i a n o f a f u n c t i o n 7?: W - * R i s a f u n c t i o n i n W O ) d e f i n e d b y t h ef o r m u l a

A%(v) = 2 - 25 ~ E ( ~ ( L ~ v ) - v ( v ) ) = 2 - 2k E Z 6 k C Z 6

I n o t h e r w o r d s ,A s _ 2 - 2

(2 .4)

k E Z 6N o t e t h a t t h e L a p l a c i a n o f x 2 r e s t r i c t e d t o V ~ i s 2 . T h a t i s t h e r e a s o n f o r t h e f a c t o r 5~2 2 .

C l e a r ly , t h e o p e r a t o r s I , L ~ , 0 ~ a n d A ~ c o m m u t e w i t h e a c h o t h e r .F o r a n y f u n c t i o n g d e f i n e d o n a s u b s e t W C _ V ~, we wi l l u s e ] ] g i i w t o d e n o t e i t s

L ~ ( W ) - n o r m :I I g l I w = s u p I g ( v ) l .y E W

F o r a n y d i f fe r e nt ia b l e f u n ct i o n G : ~ - - ~ R a n d a n y k c Z 6 , l e t O kG d e n o t e t h e d i r e c -t i o n a l d e r i v a t i v e

c3kG(z) = l i m G ( z + t w k ) - - G ( z) (2 .5)t- -*O t

D e f i n i t i o n s . L e t f : ~ - + C d b e s o m e f u n c t io n d e f in e d i n s o m e d o m a i n ~ c C . F o r e a c h~ > 0 , l e t f f b e s o m e f u n c t i o n d e f i n e d o n s o m e s e t o f v e r t i c e s V ~ c V ~ , w i t h v a l u e s i n C d .S u p p o s e t h a t f o r e a c h z E 12 t h e r e a r e s o m e 5 1 ,5 2 > 0 s u c h t h a t { v E V ~ : Iv - z I < 52 } C Vow h e n e v e r E E (0 , 5 1) .

I f fo r e v e r y z E ~ a n d e v e r y a > 0 t h e r e a r e s o m e 5 1 , 52 > 0 s u c h t h a t I f ( z ) - f ~ ( v ) I < a ,f o r e v e r y ~ E ( 0 , 5 1) a n d e v e r y v E V ~ w i t h I v - z I < 5 2 , t h e n w e s a y t h a t f ~ c o n v e r g e s t o f ,l o c a l l y u n i f o r m l y i n ~ .


6/27

224 Z . - X . H E A N D O . S C H R A M M

L e t n c N , a n d s u p p o s e t h a t f i s C ' ~ - s m o o t h . I f f o r e v e r y s e q u e n c e k l , .. ., k j E Z 6 w i t hj


7/27

C C ~ -C O N V ER G EN C E O F D ISK PA C K IN G S 225

a n d i n d u c t i o n . S i m i l a rl y , p a r t s ( 3 ) a n d ( 4 ) f o ll o w f r o m t h e i d e n t i t i e s ,- O ~ h eO ~ ( 1 / h e ) = heL~ kh~

O~v / ~ = O~he

T h e d e t ai l s a r e l e ft t o t h e r e a d e r . T h e c o n v e r g e n c e o f ]h el is C ~ , b e c a u s e I h e l = ~ . [ ]

3 . T h e s e t u pT h i s s e c t io n w i ll d e sc r i b e th e c o n s t r u c t i o n o f t h e p a c k i n g s p c . T h e n o t a t i o n s a n d a s-s u m p t i o n s i n t r o d u c e d h e r e w il l b e a d o p t e d t h r o u g h o u t t h e f o ll o w in g .

L e t ~ b e a s i m p l y - c o n n e c t e d d o m a i n i n C w i t h ~ t r F i x t w o a r b i t r a r y d is t i n c tp o i n t s z 0 , z ~ i n ~ . W e n o w d e s c r i b e a s t a n d a r d c o n s t r u c t i o n o f a t r i a n g u l a t i o n o f as i m p l y - c o n n e c t e d s u b r e g i o n o f ~ t h a t a p p r o x i m a t e s 12. F o r e v e r y s m a l l ~ > 0 , c o n s i d e rt h e s u b s e t o f v e r t i c e s o f V e A { z E C : I z l ~ 1 / ~ } s u c h t h a t t h e i r d i s t a n c e t o C - ~ i s b i g g e rt h a n E ; a n d l e t V ~ b e t h e s e t o f v e r t i c e s w h i c h a r e e i t h e r a n i n t e r i o r v e r t e x o f t h i s s u b s e t ,o r s h a re s a n e d g e w i t h a n i n t e r i o r v e r t e x . L e t V ~ b e t h e c o n n e c t e d c o m p o n e n t o f V ~w h i c h c o n t a in s s o m e v e r t e x w i t h d i s ta n c e a t m o s t ~ t o z 0, a n d l e t H ~ b e t h e s u b g r a p h o fH e s p a n n e d b y t h e v e r t i c e s i n V ~ . T h e n f o r s m a l l s , H ~ i s e q u a l t o t h e 1 - s k e l e to n o f ag e o m e t r i c t r i a n g u l a t i o n ( w i t h e q u i l a t e r a l t r i a n g l e s o f s i d e l e n g t h E ) o f a c l o s e d to p o l o g i c a ld i s k c o n t a in e d i n g t t h a t a p p r o x i m a t e s ~ .

A s w e s t a t e d i n t h e i n t r o d u c t i o n , i t f o ll o w s f r o m t h e D i s k P a c k i n g T h e o r e m t h a tt h e r e i s a d i s k p a c k i n g P ~ = ( P ~ : v E V ~ ) i n U w h o s e n e rv e is H ~ a n d w i t h t h e p r o p e r t yt h a t b o u n d a r y d i s k s a r e t a n g e n t t o O U . W e n o r m a l i z e p e b y a ( p o s s ib l y o r ie n t a t i o n -r e v e r s i n g ) M S b i u s t r a n s f o r m a t i o n , s o t h a t ( 1 ) f o r a v e r t e x v 0 E V ~ t h a t i s c l o s e s t t o z 0 t h ecen t e r o f t he d i s k P~ i s 0 , ( 2 ) f o r a ve r t e x v ~ E V ~ t h a t i s c l o s e st t o z ~ t h e c e n t e r o f th eV0d i s k P ~ i s o n t h e p o s i t i v e r e a l ra y , a n d ( 3 ) t h e s i x d is k s P ~ , k = O , 1 , . .. , 5 , s u r r ou nd P~i n p o s i t i v e c i r c u l a r o r d e r . L e t r e : V~---*U b e t h e f u n c t io n t h a t m a p s e v e r y v E V ~ t o t h ec e n t e r o f t h e d i s k P ~ . T h e n t h e R o d i n - S u l l i v a n T h e o r e m [ 15 ] t e ll s u s th a t f ~ c o n v e r g e sl o c a ll y u n i f o r m l y i n ~ t o t h e R i e m a n n m a p f : ~ t - ~ U s a t i s fy i n g f ( z 0 ) = 0 a n d f ( z ~ ) ) > 0 .

L e t R ~ d e n o t e t h e r e g u l a r h e x a g o n a l d i s k p a c k i n g w h o s e d i s k s a r e c e n t e r e d a t t h ev e r t i c e s i n t h e h e x a g o n a l g r i d H e . T h e d i s k s o f R ~ a l l h a v e r a d i u s 8 9 L e t R ~ b e t h es u b p a c k i n g c o r r e s p o n d i n g t o t h e s u b s e t o f v e r t i c e s V~ .

F o r a n y t r i p l e t o f m u t u a l l y t a n g e n t d i s k s P 1 , P 2 , P 3 i n p e o r R e , t h e r e i s a d i sk Dw h o s e b o u n d a r y cOD p a s se s t h r o u g h t h e i n t e rs e c t i o n p o i n t s P 1 A P 2 , P 2 A P 3 a n d P 3 N P 1 .


8/27

226 Z.-X. HE AND O. SCHRAM M

T h e d i s k D i s c a l l e d a dual disk o f t h e p a c k i ng . N o t e t h a t O D is o r t h o g o n a l t o e a c h o ft h e c i r c l e s O P j , j = 1, 2, 3.

4 . T h e d i sc r e t e S c h w a r z i a n s

L e t f b e a c o m p l e x a n a l y t i c f u n c t io n d e f in e d o n a d o m a i n i n t h e p l a n e C , s u c h th a t f ' ( z )n e v e r v a n i sh e s . T h e S c h w a r z i a n d e r i v a t i v e o f f i s d e f i n e d b y

i ' , , i z ) 3 1 i " l z ) ) ' ( 4 .1 )\ i f ( z ) ] 2 \ i f ( z ) ] f ' (z ) 2 ( f ' ( z ) ) 2

T h e S c h w a r z i a n d e r i v a t i v e o f f i s i t s e l f a c o m p l e x a n a l y t i c f u n c t i o n . I t is e l e m e n t a r yt o c h e c k t h a t f o r a M h b i u s t r a n s f o r m a t i o n T ( z ) = ( a z + b ) / ( c z + d ) w e h a v e 8 ( T o f ) ( z ) =S f ( z ) . M o r e o v e r , S f = O i f a n d o n l y i f f i s e q u a l t o t h e r e s t r i c t i o n o f a M h b i u s t r a n s f o r -m a t i o n . T h e s e a n d s o m e f u r t h e r p r o p e r t i e s o f t h e S c h w a r z i an c a n b e f o u n d i n [ 11 , wf o r e x a m p l e .

I n t h e s a m e s p i r i t , w e w il l d e f i n e t h e M h b i u s i n v a r i a n t s o f h e x a g o n a l d i s k p a c k i n g s ,a n d d e r i v e t h e i r i m m e d i a t e e q u a t i o n s . A n a l o g o u s i n v a r ia n t s a n d e q u a t i o n s w e r e w o r k e do u t i n [ 16 ] i n a s im i l a r w a y f o r c ir c l e p a t t e r n s b a s e d o n t h e s q u a r e g r i d , w h e r e a p p l i c a t i o n sw e r e f o u n d t o t h e s t u d y o f g l o b al p r o p e r t i e s o f i m m e r s e d p a t t e r n s i n C . H e r e , w e w i llu s e M 6 b i u s i n v a r i an t s a s a n i n t e r m e d i a t e m e a n s i n t h e s t u d y o f t h e c o n v e r g e n c e p ro b l e m ;a n d t h e S c h w a r z i a n s w i ll b e d e f i n e d a s s o m e s u i t a b l y s c a l e d m e a s u r e o f d e f o r m a t i o n o ft h e M 6 b i u s i n v a r i a n t s f r o m t h e i r r e g u l a r v a lu e s . A s a n i m p o r t a n t s t e p , w e w i ll d e r i v e af o r m u l a f o r t h e L a p l a c i a n o f t h e S c h w a r z ia n s i n t h e n e x t s e c t i o n .

O u r m e t h o d w i l l y i e l d th e s a m e r e s u l t i f R o d i n ' s e q u a t i o n ( 1 . 1 ), w h i c h t h e r a d i is a t is f y , i s u s e d i n s t e a d . T h u s , t h e u s e o f M h b i u s i n v a r i a n t s i s n o t e s s e n t i a l. W e h a v ec h o s e n t o w o r k w i t h t h e M h b i u s i n v a r i a n t s a s t h e y y i e l d r e l a t iv e l y e a s ie r e q u a t i o n s .

F o r a n y e d g e e = I v, u ] i n H ~ , w e l e t P c d e n o t e t h e p o i n t o f t a n g e n c y o f t h e t w o d i s k sP ~ , P ~ . L e t e - - I v , u ] b e s o m e e d g e i n H ~ , l e t w l , w 2 b e t h e t w o v e r t i c e s o f V ~ t h a t n e i g h -b o r w i t h b o t h v a n d u , a n d s u p p o s e t h a t w l , w2 E V ~. L e t T b e a M h b i u s t r a n s f o r m a t i o nt h a t s e n d s t h e t a n g e n c y p o i n t P c t o i n fi n i ty . T h e n t h e t w o c i rc l e s OPv , OP, , a r e m a p p e dt o l i ne s . I t f o ll o w s t h a t t h e f o u r p o i n t s T(p[v,,~l]),T(p[v,~2j),T(p[~,,~,]),T(p[~,,w2] a r e a tt h e c o r n e r s o f a r e c t a n g l e , s e e F i g u r e 4 . 1.

S i n c e T i s w e l l - d e t e r m i n e d u p t o p o s t - c o m p o s i n g b y a si m i l a ri t y , t h e a s p e c t r a t i o o ft h e r e c t a n g l e ,

IT(P[~,wI])-T(P[v,~,~])IIT(P[~,~I])-T(P [~,,~,])t '


9/27

C~176 OF DISK PACKINGS 22 7

TPw~

T P ~ T P ~

T P ~ I

Fig. 4.1. The confi guration after applying T.i s 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 . I t a l s o d o e s n o t c h a n g e i f w e m o d i f y P ~ b y a M 5 b i u st r a n s f o r m a t i o n . S e t s~ t o b e t h i s a s p e c t r a t i o d i v i d e d b y x / ~ ,

a n d l e ts ( e ) = I T ( p c , ' ( 4 . 2 )

h(e) = ~ -2 (s(e) - 1) (4.3)b e c a l l e d t h e ( d i s c re t e ) Schw arzian derivative, o r Schwarz ian o f P e a t e . T h e f a c t o r 1 / v ~in (4 .2 ) i s j u s t i f i ed b y t h e f a c t t h a t wh en t h e d i sk s P ~ , P v , P w~ , P w2 a re a l l t h e sam e s i ze ,w e g et s ( e ) = l a n d h ( e ) = 0 . T h e f a c t o r E 2 i s r e a s o na b l e , b e c au s e o f t h e b e h a v i o r o f t h eS c h w a r z i a n d e r i v a t i v e u n d e r r e s c a l i n g , n a m e l y , ( S g ) ( z ) = ~ 2 ( S f ) ( e z ) w h e n g ( z ) = f ( e z ) .( T h i s i s a l s o j u s t i f i e d b y t h e e s t i m a t e o f w

F o r a v e r t e x v i n Y ~ , d en o t e ek (v ) = [v , L~ :(v) ]. S ee F ig u re 4 .2 . Le t Sk, hk: (V~) (1) -~ Rb e d e f i n e d b y s k ( v ) = s ( e k ( v ) ) a n d h k ( v ) = h ( e k ( v ) ) . Clear ly , Sk (V)= Sk+ 3(Lkv ) a n dh k ( v ) = h k + 3 ( L k V ) .

LEM M A 4 .1 . Let v be an interior vertex in V~. Then,8 k ( V ) + S k + 2 ( V ) + S k + ( V ) = 3 8 k ( V ) S k + l ( V ) S k + 2 ( V ) (4.4)

i s val id for any k E Z6.A l t h o u g h w e w i ll n o t p r o v e i t h e r e , t h e e q u a t i o n s ( 4 .4 ) a r e s u f f i c ie n t t o g u a r a n t e e

t h a t a p o s i t i v e f u n c t i o n s o n t h e e d g e s o f H e c o r r e s p o n d s t o a n ( i m m e r s e d ) h e x a g o n a lc i r c l e p a t t e rn ( c f . [ 1 6 ] ) .


10/27

2 2 8 Z . - X . H E A N D O . S CH RA M M

L 2 v L l v\ /e 2 ( v ) e l ( v )\ /

L a v - - e a ( v ) - - v - - e o (v ) - - L o v/ \~ (v) e~ (v )

/ \L 4 v L s v

F i g . 4 .2 . T h e e d g ~ a r o u n d a v e r t e x .

q l /:q 2 _ P 2 - P l J q 0,,/ ', \ ~ \\

i\ P3 P0 ',~'~Pa P5 q5

/ \q4\\ \

F i g . 4 . 3 . T h e s p e c i a l p o i n t s o f a fl o w er .Proof. F o r a n y k E Z ~ , l et P k b e t h e p o i n t o f t a n g e n c y P~N P~%v, a n d l e t q k b e t h e

p o i n t i n P~%vnP~%+l . S e e F i g u r e 4 . 3. T h e r e i s n o l os s o f g e n e r a l it y , b e c a u s e t h e p a c k i n gP ~ m a y b e r e f l e c te d a b o u t a li n e.

L e t m k = m k ( v ) b e t h e o r i e n t a t io n - p r e s e r v in g M S b i u s t r a n s f o r m a t i o n t h a t t a k e sP k , P k - 1 , q k - 1 t o ( x ) , 0 , 1 , r e s p e c ti v e l y . T h e n , b y t h e d e f i n it i o n o f t h e sk ' s ,

mk(v)(pk+l)=--V~ski ,mk(v) (qk)=l- -v~sk i ,mk(~)(pk)=~.

C o n s e q u e n t l y , s e t t i n g M k = m k + l O m k 1, w e g e tM k (- - Vf3 Ski) = OC,

M k ( 1 - - V ~ S k i ) ~ - l ,Mk(oC ) =0.


11/27

C ~ O F D I S K P A C K I N G S 22 9

T h e r e f o r e , ( 0 i )M k = - -X / ~ S k 'w h e r e t h e u s u a l m a t r i x n o t a t i o n f o r M S b i u s tr a n s f o r m a t i o n s i s u s e d . N o t e t h a t t h e c o m -p o s i t i o n M 5 o M 4 o M 3 o M 2 o M l o M o i s t h e i d e n t i t y . H e n c e , M 5 o M a o M 3 = M o l oM ~ l o M ~ 1,w h i c h e v a l u a t e s t o

( V / ~ 8 4 3 i 8 3 8 4 - -i ) = I V ~ ( 8 0 ~ - 8 2 - 3 8 0 8 1 8 2 ) i - -3 i 8 0 8 1 ~ ( 4 . 6 )3 i s 4 s 5 -- i V ~ ( s 3 + s 5 - - 3 s 3 s 4 s 5 ) \ i - 3 i s l s 2 V ~ S l ] "

T h i s i s a n e q u a l i t y o f M S b i u s t r a n s f o r m a t i o n s , a n d i s t h e r e f o r e v a l i d u p t o a s c a l a r f a c t o r .H o w e v e r , b o t h s i d e s h a v e d e t e r m i n a n t 1 , b e c a u s e t h e m a t r i x i n ( 4 . 5 ) h a s d e t e r m i n a n t 1 ,an d t h e re fo re (4 .6 ) i s v a l i d u p t o s ig n . F ro m t h e u p p e r l e f t en t r i e s , we g e t

+ s 4 = So + s 2 - 3 S o S l S 2 . (4.7)Bec au se s4 i s n ev er ze ro , an d s i n ce t h e se t o f co n f ig u ra t i o n s o f 6 d isk s i n a ' f l o wer ' a ro u n da g i v e n d i s k is c o n n e c t e d , t h e s i g n i n ( 4 .7 ) d o e s n o t d e p e n d o n t h e c o n f i g u r a t i o n . W h e na l l c i rc l e s h a v e t h e s a m e r a d i u s , s k = l , s o t h e c o r r e c t s i g n i s m i n u s . T h i s p r o v e s ( 4 .4 ) f o rk = O . T h e e q u a t i o n s f o r t h e o t h e r v a l u e s o f k a r e v a l i d b y s y m m e t r y . [ ]

5 . T h e L a p l a c i a n o f t h e d i s c r e t e S c h w a r z i a n sI n t h i s s e c t io n , w e w i l l u s e t h e e q u a t i o n s o f t h e s k ' s o f w t o o b t a i n t h e e q u a t i o n sfo r t h e hk ' s , a n d t h e s e w i l l b e u s e d t o s h o w t h a t A ~ h k (v ) is e q u a l t o a p o l y n o m i a l in~ , h j o ( v ) , L j ~ h j l ( v ); j o , j l , j 2 E Z 6 . W e c o n s i d e r o n l y c i n t h e r a n g e ( 0, 1 ), a n d t h e r e f o r e ,i f a l l h k ' s a r e u n i f o r m l y b o u n d e d i n a v e r t e x s u b s e t W , t h e n s o a r e t h e A ~ h k ' S i ~ t h eset W (1) .

W e s u b s t i t u t e S k ( V ) = l + z 2 h k ( v ) i n eq u a t i o n (4 .4 ) , s imp l i fy , an d g e thk (v) + hk+2 (v) +hk+a (v) = 3hk ( v) + 3 h k + l ( V ) + 3 h k + 2 ( v)

S e t

+ 3 r (5.1)+ 3 ~ahk( v ) hk+ l ( v ) hk+2( v).

qYk+l(v ) = - - ( hk ( v ) hk +l ( v ) +h k+ l( v ) hk+2( v) +hk+2 (v ) h k ( v ) )_ ~ 2 h k ( v ) h k + l (v ) h k +2 (v ) .

( 5 . 2 )T h e n e q u a t i o n ( 5 . 1 ) b e c o m e s

2 h k ( V ) + 3 h k + l ( v ) + 2 h k + 2 ( v ) - - h k + a ( v ) = 3 ~2 ~k +l (V ) . ( 5 .3 )R e p l a c e k b y k + 2 , m u l t i p l y b y 2 , a d d t o ( 5 .3 ) , a n d g e t

3 h k + l ( V ) + 6 h k + 2 ( v )T 6 h k + 3 ( v )- + - 3 h k + 4 ( v ) = 3 e 2 q J k + l (V ) T 6 C 2 q 2 k + 3 ( V ) . (5.4)


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230 Z . - X . H E A N D O , S C H R A M M

1 w 12 2/ \

2 2v0 3 - - 3 Vz

2 2\ /2 2

1 u 1F i g . 5 .1 . T h e n u m b e r s m a r k t h e c o e f fi c i e n ts o f t h e h - v a l u e s i n t h e i d e n t i t i e s ( 5 . 5 ) t h r o u g h ( 5 . 8 ) .L E M M A 5 . 1 . AChk(v) is equal to a polynom ial in the variables ~ , h jo(V) , L~l h j~(v);

Jo , j l , j 2 E Z6 . M ore specifica lly ,2 ~ a e 2 ~ a ~ 2 L ~ k O k + 3 _ 2 k O k "~ h k = ~ L k + l k O k + 3+ ~ L k+ k b k + 5 + ~ L k + 5 ~ k + ~ L k + 5 ~ k + 2 - -

I t i s q u i t e f o r t u n a t e t h a t A ~ h k (v ) is a p o l y n o m i a l in r hjo(V), L~ h j2 (v ) ; j o , j l , j 2 E Z 6 ,31s i n c e t h a t s i m p l i f ie s m a n y o f t h e a r g u m e n t s t h a t f o ll o w . H o w e v e r , t h e p r o o f c o u l d b em a d e t o w o r k i f Aehk(V) w a s o n ly a C a - f u n c t i o n o f t h e s e v a r ia b l es . T h i s m i g h t b ei m p o r t a n t f o r g e n e r a l i z i n g t h e m e t h o d s o f t h i s w o r k t o o t h e r s e tt i n g s . S e e w f o r f u r t h e rd i scu ss io n .

Proof. W e w o r k o n t h e c a s e k = O . F i x s o m e vo C(V~) (2 ) . L e t v l = L o v o , u = L s v oa n d W = L l V o . T o u n d e r s t a n d t h e f o ll o w in g c o m p u t a t i o n , t h e r e a d e r i s a d v i s ed t o c o n s u l tF ig u re 5 .1 .

A p p l y e q u a t i o n ( 5. 4) w i t h v r e p l a c e d b y u a n d k r e p l a c e d b y 5, t h e n u s e t h e r e l a t i o n sh k ( v ) = h k + 3 ( L k v ) , t o g e t ( a f t e r d iv i s i o n b y 3 )

ho (u) + 2 ha (Vl) + 2 h5 (vo) + ho (L4v o) = r kO0 (u) + 2 E2kO2 (u) . ( 5 . 5 )S i m i l a r l y , a p p l y e q u a t i o n ( 5 . 4 ) w i t h v r e p l a c e d b y w a n d k r e p l a c e d b y 2 , a n d o b t a i n

h0 (L2v0) + 2 h i (vo) + 2h2 (Vl) +h 0 (w ) - - ~2~ 3 (w) +2~2k05 (w) . (5.6)Th e su b s t i t u t i o n v 0 fo r v an d 5 fo r k i n (5 .3 ) g iv es

2 h 5 ( v o ) + 3 h o ( v o ) + 2 h z ( v o ) - h o ( L 3 v o ) = 3~2kO0(v0). ( 5 . 7 )S imi l a r l y , t h e su b s t i t u t i o n v l fo r v an d 2 fo r k i n (5.3 ) g iv es

2 h 2 ( v l ) + 3 h o ( vo ) + 2 h 4 ( v l ) - h o ( v l ) = 3 E2 ~3 (v l ) . ( 5 .8 )


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C ~ O F D I S K P A C K I N G S 231

N ow sub t r ac t the su m of equa t io ns ( 5 .7) and ( 5.8) f r om the sum o f equa t ions ( 5 .5) and(5.6) , and get

3 c ~ A ~ h o ( v o ) : e 2 k O 3 ( w ) ~ - 2 E 2 ~ 5 ( w ) + e 2 k~ o ( u ) + 2 ~ 2 ~ 2 ( u ) - - a ~ 2 v ~ 3 (V l ) - - 3 c 2 ~ o ( v o ) .This p r oves the lemm a f or the case k- - 0 . For o the r va lues of k the lemm a i s a l so va l id ,b y s y m m e t r y . [ ]

6 . T h e d i s c r e t e S c h w a r z i a n s a r e b o u n d e dI n th i s s ec t ion , w e w i l l r eca l l the r esu l t o f [ 6] w hich es sen t ia l ly says tha t the d i sc r e teS c h w a r zi a n d e r i v a ti v e i s u n i f o r m l y b o u n d e d i n a n y c o m p a c t s u b s e t o f t h e d o m a i n ~ .The bo un d i s indep enden t o f e . P r ec i se ly , w e have the f o l low ing lemma.

LEMMA 6.1 . L e t vo be a ver t ex o f V ~ , a nd suppose t ha t the d i s t ance 5 f ro m vo t o{ z E C : I z l > l / ~ } - l } i s grea t e r t han 2E . T he n

I hk( vo) l =~- 21 Sk( V 0 ) - - I I ~< C, k e Z 6 , ( 6.1)f o r s o m e c o n s t a n t C = C ( 5 ) , w h i c h de p e nd s o n l y o n 5 .

T h e p r o o f is q u i t e s im i l a r t o t h e p r o o f o f L e m m a 1 .5 in [ 5 ]. T h e b o u n d e d n e s s o ft h e S c h w a rz i a n s i s e q u i v a le n t t o t h e b o u n d e d n e s s o f t h e t h i r d o r d e r d e r i v a ti v e s o f f t .I t i s a n o p e n p r o b l e m w h e t h e r t h e a b o v e l e m m a c a n b e p r o v e d d i r e c t l y u s i n g R o d i n ' sequ a t ion ( 1.1) f or the r ad i i, o r f r om the f or m ula f or the L aplac ian of the Schw ar z ians .

P roof . By [ 6] , the r e i s a homeo mo r phism g~ f r om the ca r r ie r o f R~ onto th e ca r r ie rof P~ w i th the f o l low ing pr oper t ie s :

(1) F or each v E V ~ , the im age of the d i sk R~ under gC i s the c or r espond ing d i sk P~ .( 2) T he r es t r ic t ion o f g~ to a dua l d i sk of the packing R ~ i s equa l to a M Sbius

t r a n s f o r m a t i o n .( 3) T h e r e i s a u n i v e r sa l c o n s t a n t C 1 > 1 s u c h t h a t f o r e a c h v e ( V g ) ( 2) , t h e m a p g ~

r es t r ic ted to R~ i s Cl - quas iconf or mal .(4) For each v e (V~)(2) , the re is a co nst ant C 2 - - C 2 ( 5 ( v ) ) > 0 , w h i c h d e p e n d s o n l y o n

t h e d i s t a n c e 5 ( v ) f r o m v t o { z E C : ] z ] > l / ~ } - 1 2 , s u c h t h a t t h e a r e a o f t h e s u b s e t o f R ~w her e g~ f a i l s to be conf or mal i s bounded by C2E 4. (N ote tha t the area o f R~ is 88

C o n s i d e r t h e r e s t r i c ti o n o f g~ t o R ~ o . L e t / ~ k b e t h e d u a l d i s k b o u n d e d b y t h e c i r c lew h i c h p a s s e s t h r o u g h t h e t a n g e n c y p o i n t s o f p a ir s o f t h e d i s ks Revo , RL~voE and R~k+l voSee F igur e 6 .1 .

Le t z l , z2 , z3 , z4 be a qua dr uple of po in t s on the c i r c le OR ~ o whic h are: (1) suf f icientlyspaced out , s ay , izj l_zj21>~ 7~6e1 f o r a n y j ~ j 2 ; ( 2) a w a y f ro m t h e p o i n t s o f t a n g e n c y


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232 Z.-X. HE AND O. SCHRAMM

Fig. 6.1. The six dual disks.

{Wk } : Ri o N ReLkvo, say, I j -- wkl ~> 1_ff5,19 and (3) cyclicly ordered in the counter-clockwisedirection. We claim that for any such quadruple,

liB (zl), g (z3), (z4 )]- [Zl, z2; z3, z4]l 2, (6.2)where [ . , . ; . , - ] denotes the cross rat io , and C3 depends only on 6.

In fact , there are posit ive numbers m and m* such that the quadr i laterals(R~o ;Zl, z2, z3, z4) an d (g~(R~o); ge(z l), g~ (z2), g~(z3), g~ (z4)) are confo rmal ly ho meo mor -ph ic to the s tandard r ec tang les

(Qm = [0, m] x [0,1]; (0, 0), (m , 0), (m , 1), (0,1))an d

(Qm* = [0, m*] x [0, 1]; (0, 0), (m *, 0), (m* , 1), (0, 1)),respectively. The map ge wil l then be translated to a Cl-quasiconformal map F e betweenthe s tandard r ec tang les

(Qm; (0, 0), (m, 0), (m, 1), (0, 1)) an d (Qm . ; (0, 0), (m*, 0), (m*, 1), (0,1 )).The relat ions I jl -zj 2 I>~ ~ ~, j l r imply tha t m and m*C [m/C1, mC1] are boundedfrom above and below by some universal posi t ive constants . On the oth er hand, s incei z j_wkl> TffSe1 by proper t y (2) above we deduce that g~ is conformal in the 2~0 e-neighbo rhood of the points zj, l ~j~


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C~ OF DISK PACKINGS 23 3

f o ll o w s b y a s t a n d a r d G r h t s c h a r g u m e n t ( c o m p a r e [ 6, w t h a t Im*/m-11 O .


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234 Z.-X. HE AND O. SCHR AMM

Let

w l = { x + i y 9 W o : = / > 89Since W o = R ( W o ) is contained in W, the function g=~?-~ioR is well-defined in Wo.Denote ~=x- 89 Consider the "comparison" function ~: Wo--*R defined by

[7(x +iy )=~ (~+ 89 =(7(22+y2)+la(~.-s , (7.3)and note that ~>0 on W1.

At any interior vertex of W0, we have A~(g-t))~>0 , becauseA ~ a(v ) = a ~ ( v ) - a ~ ( R v ) 1> - 2 H a ' v l l w ( , ,

and A~ (v ) =-2[[Ae~l][w(1 ) .In particular, A6(g--~)~0 at any interior vertex of W1. Then it is elementary to see

(from (2.4)) that the maximum of g-~ on W1 is attained at some boundary vertex, sayv , = x , + i y , , of W1 (the maximum principle). We claim that

( 9 - ~ ) ( v . ) ~ < 8 8 ~ ~ I 1 ~ , 7 1 1 w ( ,~ . ( 7 . 4 )Clearly, the set of boundary vertices of W1 is contained in the union of the subsets

B l = { x + i y E W l : x = 89B2 = { x + i yE W1 : x = e} ,B 3 = { x + i y E W ' : ~ ( x - ! e ' 2 * " 2 ~ l - ~ e }-~

If v , = 8 9 1 4 9 then R v , = v , , g(v,)----0, and g(v,)=O7 0 - ~ ) ~ II ,I I~~ > 7 ( 1 - ~ ) ~ l l ~ l l w / > 2 1 1 ~ l l w / > I g ( v . ) l ,

and therefore again (g-~)(v.)~0.Now let v . = e + i y . 9 In this case, E + i y . is an interior vertex of W0. Since

A~(g-~)(v.)~>0, and ( g - ~ ) ( r is the maximum of g - ~ in W1, and iy . is the onlyneighbor of v . = E + i y , outside of W1, it follows that

( g - ~ ) ( i u . ) > / ( g - ~ ) ( ~ + i y . ) .


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C ~ -C ONVER GENC E OF DIS K P AC KINGS 235

A s g ( i y . ) = g ( R ( e + i y . ) ) = - g ( ~ + i y . ) , w e d e d u c e t h a t 8 ( e + i y . ) - 8 ( i y . ) > ~ 2 g ( r a n dt h e n

(g -8 ) ( e+ iy , )


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2 3 6 Z.-X. HE AND O. SCHRAMM

h o ld s w h e n e v e r s < a , a n d k 0 , k l , . . . , k n E Z 6 . I n o t h e r w o r d s , t h e f u n c t i o n s h~k a r e u n i -f o r m l y b o u n d e d i n C ~ ( ~ t ) .

Pr o o f . T h e p r o o f w i ll b e in d u c t i v e . T h e c a s e n = 0 i s h a n d l e d b y L e m m a 6 .1 . S oa s s u m e t h a t n > 0 , a n d t h a t t h e l e m m a h o l d s f o r 0 , 1 , 2 , .. ., n - 1 . S e t

T h e n ,

0 ~ , ~ E h Eg = k,~_~ " ' - ' k l k0"

A ~ - A~ 0 ~ ~ h ~ - 0 ~ ~ A ~ h ~Y~ k,~_l""C'kl ko-- kn- l ' "Vk l ko"F r o m L e m m a 5 .1 i t n o w f o l lo w s t h a t A ~ g i s a l i n e a r c o m b i n a t i o n o f f u n c t io n s o f th e f o r m

0 ~ . _ 1 . . . 0 ~ A~j ~ ,

w i t h A = L ~ 2 o r A = I , t h e i d e n t i t y o p e r a t o r . R e c a l l , f r o m ( 5 .2 ) , t h a t ~ j i s a p o l y n o m i a li n e a n d t h e h k ' s . A l s o n o t e t h e r u l e f o r d i s c r e te d i f f e r e n t i a ti o n o f a p r o d u c t ,

w h i c h i s e a s y t o v e r i fy . F r o m t h i s r u l e , i t f o ll o w s t h a t 0 ~ 2 ~ j l i s a p o l y n o m i a l i n e a n dt h e A h j ' s , w h e r e A r a n g e s o v e r t h e o p e r a t o r s I , ~ , L j 2 . B y i n d u c t i o n , i t f o ll o w s t h a t

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

L ~ L ~ 9 e 0 ~ (8 .5)j . ~ " " j ~ + i v j ~ " '" j l h j o ,w h e r e m < . n - 1 . I f v E V ~ / 2 , m < ~ n , a n d 4 n E < 6 , t h e n v ' - L j . . .. n ~j + ~ ( v ) i s in v e V [ / 4 .T h e r e f o r e , t h e i n d u c t i v e h y p o t h e s i s w i t h ' - ~ ~ , 1- L j m . . . L j . + ~ ( v ) , n ' = s , 6 = ~ 5 a p p l i e s, a n dp r o v i d e s a b o u n d f o r (8 . 5 ) a t v . S i n c e A ~ g i s a p o l y n o m i a l i n c a n d t h e e x p r e s s i o n s o f t h ef o r m ( 8 . 5 ) w i t h m < ~ n, s < . n - 1 , i t f o ll o w s t h a t t h e r e i s a c o n s t a n t C I = C ~ ( 5 ) s u c h t h a t

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Zheng-Xu He and Oded Schramm- The C^infinity -convergence of hexagonal disk packings to the Riemann...

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