ISRAEL JOURNAL OF MATHEMATICS 146 (2005), 223-242

A BASIC INEQUALITY AND NEW CHARACTERIZATION OF WHITNEY SPHERES IN A COMPLEX SPACE FORM

BY

HAIZHONG LI*

Department of Mathematical Sciences, Tsinghua University Beijing, 100083, People's Republic of China

e-mail: hli@math.tsinghua.edu.cn

AND

L U C V R A N C K E N ~

LAMATH, ISTV 2, Campus du Mont Houy, Universitd de Valenciennes 59313 Valenciennes Cedex 9, France

e-mail: luc.vrancken@univ-valenciennes.f'r

ABSTRACT

Let N n (4c) be an n-dimensional complex space form of constant holomor-

phic sectional curvature 4c and let x: M n -+ Nn(4c) be an n-dimensional

Lagrangian submanifold in N n (4c). We prove that the following inequal-

ity always hold on Mn:

[Vh] 2 > n ~ 2 1 V l - / ~ [ 2,

where h is the second fundamenta l form and H is the mean curvature

of the submanifold. We classify all submanifolds which at every point

realize the equali ty in the above inequality. As a direct consequence of

our Theorem, we give a new character izat ion of the Whitney spheres in

a complex space form.

* Supported by a research fellowship of the Alexander von Humboldt Stiftung 2001/2002 and the Zhongdian grant of NSFC.

t Partially supported by a research fellowship of the Alexander von Humboldt Stiftung. Received November 5, 2003

223

224 H. LI AND L. VRANCKEN Isr. J. Math.

1. I n t r o d u c t i o n

Let Nn(4c) be a complete, simply connected, n-dimensional Kaehler manifold

with constant holomorphic sectional curvature 4c. When c = 0, Nn(4c) = ca;

when c > O, Nn(4c) = cpn(4c); when c < O, Nn(4c) = CH~(4c). Let x: M --+

Nn(4c) be an immersion from an n-dimensional Riemannian manifold M into

N n(4c). M is called a L a g r a n g i a n s u b m a n i f o l d if the complex structure J

of N ~ (4c) carries each tangent space of M into its corresponding normal space�9

In order to state our results, we introduce the following examples.

Example 1: W h i t n e y s p h e r e in C n (see [18], [1], [3]). It can be defined as

the Lagrangian immersion of the unit sphere S n, centered at the origin of R n+l ,

in C n, given by (up to translation and scaling)

(1�9 1

, 2 ( x l , x l xn+l , " "" 'Xn'XnXn+l)�9 O : S ~ - ~ C ~, ~ ( x l , . . . X ~ + l ) - l + x n + 1

From a Riemannian point of view, this Lagrangian sphere plays the role of the

round sphere in the Lagrangian setting.

Example 2: W h i t n e y s p h e r e s in CP~(4) (see [2], [4], [9])�9 They are a one-

parameter family of Lagrangian spheres in Cpn(4), given by

O0:S ~--+CP~(4) , 0 > 0 ,

( x , , ,xn),0co(l+ xL1) + iXn+ (1.2) ( I ) ( X l , . . . II 0

CO + iSOXn+l C~ 2 2 ]' ' ..~ 80Xn+ 1

where co = cosh0, so = sinh0, II: S 2~+1 --+ c p n ( 4 ) is the Hopf projection.

We notice that O0 are embeddings except in double points, and that ~o is the

totally geodesic Lagrangian immersion of S n in cpn(4 ) .

Example 3: W h i t n e y s p h e r e s in C H n ( - 4 ) (see [2], [4], [9]). They are a

one-parameter family of Lagrangian spheres in C H ~ ( - 4 ) , given by

~e: S n --+ C H n ( - 4 ) , 0 > 0,

(1.3) Oo(xl, X n + l ) = I - i ~ ( x l ' ' ' ' ' x n ) ' s O c O ( 1 - + x 2 + l ) - - i x n + l ~ �9 ' ' , ' ~ 2 - - - 2 - 2 ] ' \88 "~ iCOXn+l ~0 T C O ~ n + 1

where co = coshS, so = sinh0, II: H~ n+l --~ C H n ( - 4 ) is the Hopf projection;

Oe are also embeddings except in double points.

2 2 - 1} Example4: I f R H n-~ = {y = (y~ , . . . , yn ) e R n : y2 + ' ' ' + yn_~ - y n =

denotes the (n - 1)-dimensional real hyperbolic space, following [2] (cf. [9]), we

Vol. 146, 2005 WHITNEY SPHERES IN A COMPLEX SPACE FORM 225

define a one-family of Lagrangian embeddings

~ : $1 • RHn-1 ~ C H ~ ( - 4 ) , fl �9 (0,7r/4],

given by

( 1 t (cos/~ cos ) (1.4) q~(eU,y) = II o t - isinflsint;y) sin fl cos t + i cos fl sin

where H: H12~+l --+ CHn(-4) is the Hopf projection.

Example 5: Following [2] (cf. [9]), we define a one-family of Lagrangian

embeddings ~)v: R n = R 1 x R n - 1 ----} C H n ( - 4 ) , p > 0,

given by

(1.5) r 1 ( 2 2 (~(L '2+t 2) +2lx[ 2 + ig~ t )

1 1 where el = 3 ( 0 , . . . , 0 , 1 , - 1 ) , e~ = ~(0 , . . . , 0 ,1 ,1 ) .

In [5] and [18], and for any Lagrangian submanifold of the complex Euclidean space C n, the complex projective space CI? n or the complex hyperbolic space

CIE n , the following universal inequality was obtained:

[h[ ~ > 3n2 [H[2, - n + 2

where h is the second fundamental form and H is the mean curvature vector.

Moreover, it was shown that a Lagrangian submanifold realizes at every point the equality in the above inequality if and only if it is totally geodesic or one of the above examples.

In this paper, we prove the following result.

Let x: M --+ Nn(4c) be an n-dimensional Lagrangian sub- MAIN THEOREM:

manifold. Then

(1.6) [X~hl 2 > ~3n--lV• - n + 2

where h is the second fundamental form and H is the mean curvature vector

of the submanifold. Moreover, the equality holds at every point in (1.6) if and

only if either k* (t) M has parallel second fundamental form, i.e., hiy,~ = 0,1 < i , j , k , l <_ n;

or

226 H. LI AND L. VRANCKEN Isr. J. Math.

(2.1) in case c = O, x( M) is an open portion of the Whitney sphere in C n, given

by (1.1);

(2.2) in case c = 4, x(M) is an open portion of one of the Whitney spheres in

cpn(4) , given by (1.2);

(2.3) in case c = -4 , x (M) is an open portion of one of the Lagrangian sub-

manifolds in CHn(-4 ) , given by (i.3), (1.4) and (1.5).

As a direct consequence of our Main Theorem, we get the following new

characterization of Whitney spheres.

COROLLARY: Let x: M -4 N~(4c) be an n-dimensional compact Lagrangian

submanifold with non-parallel mean curvature vector. Then

3n 2 I hl 2 = IV• 2

n + 2

if and only if:

(1) In case c = O, x(M) is the Whitney sphere in C n, given by (1.1).

(2) In case c = 4, x (M) is one of the Whitney spheres in cpn(4) , given by

(1.2). (3) In case c = -4 , x (M) is one of the Whitney spheres in C H n ( - 4 ) , given

by (1.3).

Remark 1.1: In [11], [12], [13], the authors established the similar inequality

(1.6) for n-dimensional submanifolds in an (n +p)-dimensional unit sphere S n+p

in different contexts.

2. Prel iminaries

Let Nn(4c) be a complete, simply connected, n-dimensional Kaehler manifold

with constant holomorphic sectional curvature 4c. Let M be an n-dimensional

Lagrangian submanifold in N n (4c). We denote the Levi-Civita connection of M

and Nn(4c) by V and XT, respectively: The formulas of Gauss and Weingarten

are respectively given by

(2.1) f T x Y = V x Y + h ( X , Y ) and f T x ~ = - A ~ X + V ~ c ~ ,

for tangent vector fields X, Y and normal vector field ~, where V • is the

connection on the normal bundle. The second fundamental form h is related to

A~ by

(2.2) < h ( X , Y ) , ~ > = < A ~ X , Y > .

Vol. 146, 2005 W H I T N E Y SPHERES IN A C O M P L E X SPACE F O R M 227

The mean curvature v e c t o r / t of M is defined b y / 7 = ~ trace h and the mean

curvature function H is the length o f / t .

For Lagrangian submanifolds, we have (cf. [8])

(2.3)

(2.4)

V~cJY : J V x Y ,

A j x Y = - J h ( X , Y) = A j y X .

The above formulas immediately imply that < h ( X , Y ) , J Z > is totally

symmetric, i.e.,

(2.5) < h(X, Y), JZ > = < h(Y, Z), J X > = < h(Z, X), JY > .

For a Lagrangian submanifold M in N n (4c), an orthonormal frame field

e l ~ . . . ~en~el*~ . . �9 ~en*

is called an adapted Lagrangian f r a m e field if el , . . . ,en are orthonormal

tangent vector fields and el*,. �9 �9 en* are normal vector fields given by

(2.6) e l . : J e l ~ . . . , e n * -- f len .

Their dual frame fields are 01 , . . . , 0n, the Levi-Civita connection forms, and

normal connection forms are Oij and 0i*j,, respectively.

Writing h(ei, ej) = ~ k hk~ ek*, (2.5) is equivalent to

k* i* j* ( 2 . 5 ) ' h i j = hk j = hik , l < i , j , k < n .

If we denote the components of curvature tensors of V and V • by R i j k l and

R~Zij, respectively, then the equations of Gauss, Codazzi and Ricci are given

by

(2.7)

(2.8)

(2.9)

j * j * __ Rmi lp -': C(~ml(~ip - ~mp~il) § E ( h m l h i p ,hJ*~mp,hJ*),~il

J k* hk~l=hil,j , l <_ i , j , k , l<n ,

X-"r~i* ~J* i* j* Ri*j*k l ---- C(SjlSik -- 5 jkSi l ) + ff_.~\,~mk,~ml -- h m l h m k ),

m

where hik-[l is defined by

(2.10) k. k" Eh ;e.§ "

l l l m

228 H. LI AND L. VRANCKEN Isr. J. Math.

We can write (2.10) in the following equivalent form:

(2.10)' (Vxh) (Y ,Z) = V~h(Y ,Z ) - h (VxY, Z) - h(Y, V x Z ) ,

where X , Y , Z are tangent vector fields on M. We note that (Vekh)(ei,ej) =

k* Combining (2.5)' with (2.8), we know hij,l totally symmetric, i.e.,

(2.11) hk~l i* = h j" = h l• ---- hj l ,k lk,i ki , j , 1 <_ i , j , k , l <_ n.

We also have the following formulas:

(2.12) hkf, tP - hk~p I = E h~jR.~i~p + E h~:Rmjl. + E h~j*R~*k*Ip, m m

(2.13) Rm*i*~p = Rmitp,

where hki~z; is defined by

(2.14) k* k* P*

P P P P P

Letting i = j in (2.10) and carrying out summation over i, we have

(2.15) E H f O, = dH a• + E Ht'Ot'k*' l l

1 ~-~i h~/*. Moreover, as a consequence of (2.11), we have that where H k* H k* I* ,l : - H , k "

3. S o m e l e m m a s

We start with the following lemmas

LEMMA 3.1 (see Montiel-Urbano [14]): Let M be an n-dimensional Lagrangian

submanifold in Nn(4c). If p is a point of M, Sp the unit sphere in TpM and

f: Sp --+ R the function given by

f(v) =< h(v,v), Jv >,

then there exists an orthonormal basis {e l , . . . , e~} of TpM satisfying

(i) h(el, el) -- AiJei, i -- 1 , . . . , n , where A1 is the maximum o f f ;

(ii) A1 _> 2A~,i = 2 , . . . , n , and if A1 = 2Aj /'or some j C {2 , . . . , n} , then

f(e ) : 0 .

Vol. 146, 2 0 0 5 WHITNEY SPHERES IN A COMPLEX SPACE FORM 229

Proof'. Let el be a vector of Sp where f attains its maximum. Then for any

unit vector v orthogonal to el, we have

(3.1) 0 = dYe 1 (V) ---- 3 < h ( e l , e l ) , J v >

and

(3.2) 0 >_ d2fel(v,v) = 6 < h(v,v), Jel > - 3 f ( e l ) .

From (3.1), we obtain that h(el, el) = AiJel , where A1 = f (e l ) . Using (2.4),

this implies that el is an eigenvector of A J e ~ . So we can choose an orthonormal

basis { e l , . . . , en} of TpM which diagonalizes Aje~, i.e., Agelei = Aiei. So using

(2.4) we prove (i).

Now, using (3.2) one has that A1 >_ 2Ai for i E { 2 , . . . , n } . If A1 = 2Aj, for

some j C { 2 , . . . , n } , then d2fe~(ej,ej) = 0, and so dUfe~(ej,ej,ej) = 0. But

using (3.1), dUfe~(ej,ej,ej) = 6f(ej). This proves (ii). |

When working at a point p of M, we will always assume that an orthonormal

basis is chosen such that Lemma 3.1 is satisfied.

Let x: M --+ N'~(4c) be an n-dimensional Lagrangian submanifold. LEMMA 3.2:

Then

3n2 iVl~l 2 (3.3) I~h12 > n + 2 '

where I~'hl 2 k* 2 = ~i , j ,k , t (hi j , t ) , ] V • = ~k , i (H,~*) ~. Moreover, the equali ty

holds in (3.3) if and only if

(3.4)

Proof:

(3.5)

It is easy to check that

(3 .6 ) 0 < IWl 2 := ~ ( w ~ ) 2 = iVhl 2 _ _ _ i , j ,k,l

k* k* hij, _ re +n 2 (H k* 5jl + H,j 5~1 + H,t 5~j), 1 _< i, j, k, l _< n.

We construct a tensor W by

k* k* W~; := hij,lk* _ ~-~"(Hk'i*(~Jtn + z + H,j 5it + H,t 5ij).

3n2 • ~ 2 r ~ 2 1 v H I ,

where IVhl 2 k* 2 -~ ~i,j,k,l(hij,l) and IV• = ~k,i(H,ik* )2. Equality holds in (3.3) if and only if ]Wl 2 -- 0, i.e., w~k~ = 0, 1 < i , j , k , l <_ n,

which is equivalent to (3.4). |

230 H. L I A N D L. V R A N C K E N Isr . J . M a t h .

LEMMA 3.3: Let x: M ~ N n ( 4c) be an n-dimensional Lagrangian submanifold. I f (3.4) holds, then

i* (3.7) H,j : ,~(~ij

for some function A on M , i.e., the vector field - J H = ~ k Hk* ek is a conformal

vector field.

Proof'. From (3.4), we have for all i, j , k, l

(3.8) hk;, l _ n 2 ( H f h j L + Hkj.5, l + H~*hij), r e + ' '

l* re l* l* 4" - (H,i 5jk + H j ~ik "b H k 5ij). (3.9) hij'k n + 2 ' '

k* From (2.11), we have h~,k = hij,l , therefore (3.8) and (3.9) imply

k* k* k* (3.10) H i, 5j lq-H,j 5 i l + H l, 5ij -- Hl;hjk, +Hl;h ik q-Hl;(~ij,,. 1 _< i , j , k , l _< n.

Taking i = 1 in (3.10) and summing over i, we have

k* 1 n

l

According to the nota t ion of [18], we note tha t the vector field - J / - I = ~-~-k Hk* ek is a conformal vector field. |

LEMMA 3.4: Let x: M --+ Nn(4c) be an n-dimensional Lagrangian submanifold.

I f (3.4) holds, then

(3.11) hk~l = #(hkihjl + 5kjhit + 5klhij),

(3.12) e~(#) = 0,

1 1 < i , j , k , 1 < re, # = n + 2 - p

/ : 2 , . . . , n ,

(3.13) - e l (p ) + (2Al - A1)(c+ AIA1 - A~) = 0, 1 = 2 , . . . , n .

I f Ajepe~ ~ 0 for s o m e p , l, 2 <_ p ~ l ~ n, we have

(3.14) e l (#) = 0,

in which case (3.12) and (3.13) imply that # is a constant.

Proof: The first s ta tement (3.11) follows immdiate ly from (3.7) and (3.4).

Taking the covariant derivative of (3.11) implies tha t

k* (3.15) hij,l p : ep(#)(hkihj l + ~kjhil -b ~kl~ij).

Vol. 146, 2005 W H I T N E Y S P H E R E S IN A C O M P L E X SPACE F O R M 231

Exchanging I with p in (3.15), we have

(3.16) h~p l = el (#)(hk~hj~ + 5kjhi~ + 5k~5ij).

Putting (2.7), (2.13), (2.9) into (2.12), we have

k* k* k* m*

m m m

k* k* = e(~ 13 ~ + ~. ~ - ~pj~" ~ - ~ ~) (3.17) + e(h~j~kp P* ~ ]~k* hn* l~n* ~ ~k* ]~n* ~n* * -- hi j (~kl) T ff_~ ,Ojm,Olm,~ip -- ~ ,~jm,Omp,~il

m,n m,n

E ~k* ~n* ~n*

m,n m,n

m* n* n* m* n* n* + E hij hm~hkp - E hij hmp hkl" m,n m,n

Substituting (3.15) and (3.16) into (3.17), we find that

k* -----c(hlj (~ip "[- h~( (~jp -- k*

(3.18) + c(h~}hkp - h~; 5kt) + ~ '~176176 ~ * ~ " _ ~ h k * hn* ~n* Z...,

m,n m,n

k* n* n* k* n* r~* + E h i m h l m h j p - E h i m h m p h j l

m~n m,n

m* n* n* n* n*

m~n m~n

1" Now we take an orthonormal frame as in Lemma 3.1. Then we have hij )~iSij,1 <_ i , j <_ n. Choosing i = j = p = 1,l r 1 in (3.18), we get that

e ~ ( ~ ) h k l -- 3 e l ( ~ ) h k ~ (3.19)

=c(2$l - ~1)5k, + ( - - 2 ~ l + 3 ~ 1 ~ -- )~.~klhkt, 1 < k < n,l # 1.

Choosing k = 1 , / r 1 in (3.19), we have

(3.20) el(#) = 0, l = 2 , . . . , n .

Choosing k = l r 1 in (3.19), we obtain (3.13). Finally, choosing i = j = 1,p

l ,p,l r 1 in (3.18), by use of (3.20) we find that

(3.21) (~p - ~l)(2~k -- A1)h~ = 0, p # l, p , l # 1,

232 H. LI AND L. VRANCKEN Isr. J. Math.

which is equivalent to

(3 .21) ' (Ap-At)(AiAj%el-2AjelAj%el)=O, p # l , p , l # l .

If Ap # Al(p,l # 1), then Ajepel C V(A1/2) (the eigenspace of Agel with

respect to eigenvalue A1/2). If Ajepel # 0 for some p, l # 1,p # l, we get (3.14)

from (3.21)' and (3.13). In this case, (3.12) and (3.14) imply that # is constant.

1

Now we first assume # #constant and therefore el(#) # 0. From (3.13), we

get

(3.22) el(#) = (2Al- / \ l ) (C + AtA1- A~), l = 2 , . . . , n .

Let y := AI - 2Al,l # 1; we have y _> 0 from Lemma 3.1. Thus we get from

(3.22)

(3.23) el(p) = Y(y2 _ 4 c - A2), y > 0.

(i) If - 4 c - A~ > 0, (3.23) has only one solution Yx > 0 and el (~) > 0.

(ii) If - 4 c - A 2 < 0 and el (#) > 0, (3.23) has only one solution Yl > 0.

(iii) If - 4 c - A~ < 0 and el (#) < 0, (3.23) has two solutions Yl > 0 and Y2 > 0.

Therefore, from the definition of y and the above analysis, we conclude that

the solutions of (3.22) satisfy one of the following two cases:

CASE i:

(3.24) A2 = A3 . . . . . An.

CASE 2:

(3.25) A2 . . . . - At-4-1 # At+2 . . . . . An.

Now we discuss Case 2 first and use the following convention about indices:

(3.26) 2 < _ a , b , c < r + l , r + 1 _< a,/3,7_< n.

LEMMA 3.5: Let x: M --+ Nn(4c) be an n-dimensional Lagrangian submanifold.

Assume that (3.11) holds and # is not a constant. In Case 2 (i.e., when (3.25)

holds), we have

(3.27) Ajeo ea = Aje~ea = O, 2 < a < r + l, r + 2 < a < n .

Vol. 146, 2 0 0 5 WHITNEY SPHERES IN A COMPLEX SPACE FORM 233

Proof: Choosing p = a, l = (~ in (3.21)', we have

(An - Aa)(A1Aje~ea - 2AJelAde~ea) -- O, 2 < a < r + 1, r + 2 < a < n.

Thus we get Aje~ea E V(A1/2) (the eigenspace of Age1 with respect to eigen-

value A1/2). If Aje~ea # 0 for some a , a , from Lemma 3.4, we conclude

el(#) = 0. Combining with (3.20), we get # is constant, which is a contra-

diction to our assumption. Thus we prove (3.27). |

LEMMA 3.6: Under the same assumptions as in Lemma 3.5, we have

(3.28) < h(U, V), J W > = O, U, V, W E V()~2)

where V(~2) is the eigenspace of Ajel with respect to eigenvalue )~2.

Proof." Let V(A2) be the eigenspace of Ajel with respect to A2. We may choose

e2 such tha t at the vector e2 the function ](v) : = < h(v,v) , Jv >, < v ,v >= 1,

restricted to V(A2) at ta ins its maximal value. Let

v = c o s t e 2 + s i n t e i , 0 < t < 2 7 r , 3 < i < r + l .

We have the function

g(t) := f(coste2 + sintei) .

It is easy to check tha t g'(0) = 0 is equivalent to

(3.29) <h(e2, e2) ,Je i>=<Age2e2, e i>=O, 3 < i < r + l .

From (3.29), we can therefore assume that

(3.30) Aj~2e2 = A2el + k2e2.

Choosing now p = k = 1,/ = i = j = 2 in (3.18), and using (3.20), (3.30)

together with Aje~ei = Aiei, we get

(3.31) (-A22 + c + ~1~2)k2 = 0.

Choosing 1 = 2 in (3.13), it follows that

(3 .32) e l ( , ) -- (2/~ 2 - / ~ 1 ) ( c ~- )tl/~ 2 - ,~22).

Since we assumed tha t e l(#) # 0, (3.32) implies that c + ~1~2 - A2 # 0. Thus

it follows from (3.31) tha t

(3.33) k2 = 0.

234 H. LI AND L. VRANCKEN Isr. J. Math.

However, (3.30) and (3.33) then imply that

(3.34) f(e2) = < h(e2, e2), Je2 > = 0.

Noting that f(e2) is the maximum value of f(v) on V(A2) and f(v) is an odd

function, we conclude that

(3.35) f ( X ) =< h ( X , X ) , J X >= 0, VX e V(A2).

For any U, V, W E V(A2), letting X = aU + bV + cW, a, b, c E R, we have

f (aU + bV + cW) = < h(aU + bV + cW, aU + bV + cW), J(aU + bV + cW) >

=0, a, b, c E R.

From the arbitrariness of a, b, c, we get (3.28). |

Using a similar argument as in the proof of Lemma 3.6, we can prove

LEMMA 3.7: Under the same assumptions as in Lemma 3.5, we have

(3.36) < h(U, V), J W > = 0, U, V, W E V(An)

where V(A,~) is the eigenspace o[ Ajel with respect to eigenvalue An.

Remark that Lemma 3.6 also remains valid in Case 1, provided # is not a

constant.

LEMMA 3.8: Under the same assumptions as in Lemma 3.5, we have

(3.37) < h(ei,ej),Jek > = 0, 2 _< i , j , k <_ n.

Proos

(i) I f i = a , j = b , k = c o r i = a , j = / 3 , k = % w e g e t (3.37) from (3.28) or

(3.36).

(ii) If i = a, j = b, k = a, we have from (2.2) and (3.27)

< h(ea,eb),Jea > = < Ageaea,eb > = 0.

(iii) If i = a, j =/3, k = a, we have from (2.2) and (3.27)

< h(ea, e~), Jea > = < Age. ea, e~ >-- O. |

As the function # is globally defined and as el(#) r 0, we see that the vector

el is characterized as the normalised dual vector to the 1-form d#. This shows

that we can extend the vector el differentiably in a neighborhood of the point

Vol. 146, 2005 WHITNEY SPHERES IN A COMPLEX SPACE FORM 235

p in such a way that at every point the function f attains a maximum at the

point p. As a consequence, the previous lemmas remain valid in a neighborhood

of the point p. We will denote the extensions of the vectors ei to vector fields,

and of the eigenvalues Ai to eigenfunctions, by using the same letters.

LEMMA 3.9: Under the same assumpt ions as in L e m m a 3.5, let

A j e l e i = Aiei, A2 . . . . . Ar+l # Ar+2 . . . . . An;

then

(3.38) A1 - 3Aa iS constant, 2 < a < r + 1,

and

(3.39) A a - A n is constant, 2 < a < r + 1 , r + 2 < a _ < n .

P r o d : From (2.11), assumption condition (3.4) is equivalent to

k* _ n j* k* (3.40) hij'l n + 2 (H~I* (~jk -~ H l (~ik ~- H,I (~ij) : 0, 1 < i, j , k, 1 < n.

Choosing i = j = k = 1 in (3.40), we have

1" n 1" (3.41) hll,t - 3 ~ - - ~ H , l = 0,

Choosing k = 1,i = j _> 2 in (3.40), we have

1" n H 1 . (3.42) h ~ i , z - ~ ,l = 0 , 2 < i < n , n +

Thus we have from (3.41) and (3.42)

(3.43) h 1. - 3 h ~ l = 0 , l < l < n . 115

From definition (2.10) and (3.37), we have h~[,l =

i = 2 , . . . , n, 1 < I < n. Thus we have from (3.43)

(3.44) et(A, - 3Ai) = 0, i = 2 , . . . , n ,

Therefore, we get from (3.44)

(3.45)

A1-3Aa is constant, A1-3A~ is constant,

We prove (3.38) and (3.39) from (3.45). |

l < / < n .

l < / < n .

1" el(A1), hii,l = ez(Ai),

l < / < n .

2 < a < r + 1 , r + l < ~ < n .

236 H. LI A N D L, V R A N C K E N Isr. J . M a t h .

LEMMA 3.10: Under the same assumptions as in Lemma 3.5, we get that

H =constant and ~Th = O, i.e., the second fundamental form h is parallel.

Proof: If H is not constant , it follows from the previous lemma tha t

(3.46) Aa is not a constant , i.e., dAa # O.

Choosing now l = a and l = a , in (3.22), respectively, we have

(3.47) (2Aa -- ~ I ) ( C "~- ,'~a,~l - - )~2a) = (2)~a -- /~ I ) (C -[- /~a,~l - - /~2) .

By (3.39), we can int roduce a constant K by

(3.48) K := Am - Aa = constant .

Pu t t ing (3.48) into (3.47), we get

(3.49) K[2c - 6A2a - 6KAa - 2K 2 + 6AIAa + 3KA1 - A12] = 0.

(3.48) and assumption condit ion (3.25) imply K # 0, thus we have

(3.50) 2c - 6A~ - 6KAa - 2K 2 + 6A1Aa + 3KA1 - A T = 0.

Differentiating (3.50) and using (3.38), we have

(3.51) ( 2 A a + K ) d A a = 0 , a = 2 , . . . , r + l ,

which is a contradic t ion with (3.46). Thus H =cons tan t .

Since H is constant , we have from Lemma 3.1 and L emma 3.8 tha t

E l ( E h i k * k J e i ) ] l J e l ' = 1 h(ek,ek) n ik " k

thus

(3.52) H 1. = H, H i* = 0, i = 2 , . . . , n .

From H is constant and the definition ~ k HI,[ 8k = dHa* + ~ k Hk* Ok*l*, we

have

1" (3.53) H,1 = 0.

Combining (3.53) with (3.7), we have A = 0, i.e., H k* = 0. We conclude now

from (3.4) t ha t h~,k = 0, i.e., the second fundamenta l form h of M is parallel. |

Vol. 146, 2005 WHITNEY SPHERES IN A COMPLEX SPACE FORM 237

PROPOSITION 3.1: Let x: M --4 N'~(4c) be an n-dimensional Lagrangian sub-

manifold. If (3.11) holds with # =constant, then # = 0 and M is of parallel

second fundamental form.

Proof:

(3.54)

By the definition of #, (3.11) and (2.10)', we have

< (Vxh)(Y, Z), J W >

=#(< Y ,Z >< X , W > + < X , Z >< Y , W > + < Y , X >< Z , W >).

At the point p E M, we choose a frame {el , . . . , en} as before such that

Aje~e~ = Ale1, Ajelei = )~iei, < h(ei,ej),Jek >= O, i , j , k > 2.

Then Jell I/4. We take a geodesic 7(s) passing through p in the direction of el. Let {El , . . . , En} be a parallel vector field along this geodesic 7(s), such that

Ei(p) = ei and E1 = ~/'(s). Then we have by use of (3.54)

0 < Ei ,Ej >= 5ij, -~s < h(E1,E1),JEi > =< (VE~h)(E1,E1),JEi >

=0, i_>2,

and

0 0--~ < h(E1, Ei), JEj > = < (VEIh)(E1, Ei), JEj >= 0,

Thus we have

i # j k 2 .

<h(E1 ,E1) , JE i > = < h ( e l , e l ) , J e i > = 0 , i > 2 ,

and

< h(E1,Ei), JEj > = < h(el,ei), Jej >= 0,

that is, we can write

i , j > 2 ,

(3.55) AjE1E1 = ~1E1, AjEIEi = ~iEi, i ~_ 2.

By use of Ricci identities and the fact that # =constant, repeating the arguments

of the proof of (3.13) we can get that along 7(s) we have

(3.56) (~1 -- 2~i)(C -- ~2 + ~i~1) : 0, i ~ 2.

However, using (3.54), we have that along 7(s)

0 - 0 (3.57) ~ss ~l(s) = ~ss < h(E1, El), JE1 > = < (VElh)(E1,E1), JE1 >= 3#

238 H. LI AND L. VRANCKEN Isr. J. Math.

and

(3.58) o 0

Ai(s) = -~s < h(Ex,Ei) ,JEi > = < (VElh)(E1,Ei) ,JEi > = #, i _> 2.

By use of (3.57) and (3.58), taking the derivative of (3.56) along "/(s) implies

that

(3.59) , ( c - 3 ~ + X~) = 0.

By use of (3.57) and (3.58), the first and second derivatives of (3.59) imply

6#2(A1 -/~i) = O, i _> 2, 12# 3 = O,

k* from which we conclude that # = 0. From (3.11) we know that hij,l ---- 0, i.e.,

M is of parallel second fundamental form. |

4. P r o o f o f M a i n T h e o r e m

From the discussions of Section 3, it follows that

Let x: M --+ Nn(4c) be an n-dimensional Lagrangian sub- PROPOSITION 4.1:

manifold; then

(4.1) IVhl 2 > 3n2 IV• - n + 2

whereas the equality holds in (4.1) at every point if and only if one of the

following two cases occurs: (i) M is of parallel second fundamental form;

(ii) for every point p belonging to an open dense subset of M there exists an

adapted Lagrangian frame field e l , . . . , en, e l* , . . . ,en* with el. parallel to H such that the second fundamental form of M in Nn(4c) takes the

following form:

h ( e l , e l ) = ~ l e . , h(e~,e~) . . . . . h(e~,en) = ~ e ~ . , (4.2)

h(e l ,e j )=A2ej . , h(ej ,ek)=O, 2 < j ~ k < _ n ,

with (4.3)

( ~ 7 x h ) ( Y , Z ) = # ( < Y , Z > J X + < X , Z > J Y + < X , Y > JZ), # r

and d# vanishes nowhere.

Vol. 146, 2 0 0 5 WHITNEY SPHERES IN A COMPLEX SPACE FORM 239

Remark 4.1: B. Y. Chen [7] called Lagrangian submanifolds with (4.2) Lagrangian H-umbilical submanifolds.

Note that the first case has been classified by H. Naitoh (see [15], [16] and

[17]). Now we discuss what happens in the second case. Note that as # is not a

constant, we must have A1 ~ 2A2. As d# vanishes nowhere, everything can be

locally extended as indicated in the previous section. Choosing X = Y = Z = el

and X = ei(i > 2), Y = Z = el in (4.3), respectively, we get that

(4.4) (~el h)(el, el) = 3#Je l

and

(4.5) (Ve~h)(e l ,e l ) = #Je i , i >_ 2.

It is a direct check using (2.10)' and (4.2), (4.4) and (4.5) that this implies

(4.6) el(A1) = 3#, V e l e l -~ 0,

and

(4.7) ei(A1) 0, Ve~el # = - - - e l , i > 2. A1 -- 2A2

Now choosing X = Y = Z = el, i > 2 in (4.3), we have

(4.8) (V~h)(e~,e i ) = 3 # J e , i _> 2.

From (2.10)' and (4.2), we get by use of (4.7)

(4.9)

((Ye~ h)(ei, ei) = V ~ h(ei, ei) - 2h(Ve~ ei, ei)

=V~(A2Jel ) - 2 < el,Ve~ei > h(e l ,e i )

- 2 ~ < ez,Ve~ei > h(et,e~) 1>2

A2# Je~ - 2 < el,Ve~e~ > A2Jei =ei(A2)Jel + A1 -2A-~

=ei(A2)Jel + 3A2# Jei , i > 2. A1 - 2A2)

It follows from (4.8) and (4.9) that

A2# p - A1 - 2A2"

240 H. LI AND L. V R A N C K E N

As # # 0, we have

(4.10) A1 = 3A2.

By use of (4.10), we now can easily check that in this case

hikj. _ n 2(Hk.5~ J �9 n + + Hi" 5kj + H ~ 5ik),

which is equivalent to

2(n + 2) n + 2 H 2 = n - ~ ( n - - 1 ) R - n c,

Isr. J. Math .

where R is the scalar curvature of M. In case c = 0, by [18], [1] or [3], x(M) is

an open portion of the Whitney sphere in C n, given by (1.1). In case c = 4, by

[9], [2], [4], x(M) is an open portion of one of the Whitney spheres in Cpn(4), given by (1.2). In case c = -4 , by [9], [2], [4], x(M) is an open portion of one of

the Lagrangian submanifolds in C H n ( - 4 ) , given by (1.3), (1.4) and (1.5). This

completes the proof of the Main Theorem.

5. R e m a r k s

From (2.11), we have

k* t" Hk* i* (5 .1 ) hij,l = hij,k , ,~ : H,k.

Thus (3.4) is equivalent to

j. t* n i*hjl + H,k 5il + H~hij) , (5.2) hij'k -- n + 2 (H'k

Define L~jl and its covariant derivative Lijl,k as follows:

n i* l* (5.3) L~jl = h~ - - - - - ~ ( H 5jt + HJ*hiz + H 5ij), n +

l < i , j , k , l < n.

l <_i,j,l <_n,

(5.4) E nijl,kOk : E nkjlOki + E LiklOkJ ~ E LijkOkl" k k k k

Thus (5.2) is equivalent t o Lijl, k = 0; this implies that (3.4) is equivalent to

Lijt,k =0, 1 < i, j, k, l <_ n. From the proof of our Main Theorem, we get

Vol. 146, 2 0 0 5 WHITNEY SPHERES IN A COMPLEX SPACE FORM 241

THEOREM 5.1: Let x: M -4 Nn(4c) be an n-dimensional Lagrangian sub-

manifold. I f

(5.5) Lijl,k =O, l <_i , j ,k , l < n,

then either

(1) M has parallel second fundamental form, i.e., h~],l = O, 1 <_ i , j , k, l <_ n;

or

(5.6) Lijl = O, 1 <_ i , j , l <_ n.

In the latter case, we have the following classifications.

(2.1) In case c = O, x (M) is an open portion of the Whitney sphere in C n, given

by (1.1).

(2.2) In case c = 4, x (M) is an open portion of one of the Whitney spheres in

c pn ( 4 ) , given by (1.2).

(2.3) /n case c = -4 , x (M) is an open portion of one of the Lagrangian sub-

manifolds in C H ~ ( - 4 ) , given by (1.3), (1.4) and (1.5).

ACKNOWLEDGEMENT: The authors express their thanks to Udo Simon for his

help.

R e f e r e n c e s

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242 H. LI AND L. VRANCKEN Isr. J. Math.

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[9] B. Y. Chen and L. Vrancken, Lagrangian submanifolds satisfying a basic equality,

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