A basis inequality and new characterization of Whitney spheres in a complex space form

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: [email protected] AND LUC VRANCKEN ~

LAMATH, I S T V 2, Campus du Mont Houy, Universitd de Valenciennes 59313 Valenciennes Cedex 9, France e-mail: [email protected]'r

ABSTRACT Let N n (4c) be an n-dimensional complex space form of constant holomorphic 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 inequality always hold on Mn:

[Vh] 2 > n ~ 2 1 V l - / ~ [ 2, where h is the second fundamental form and H is the m e a n curvature of the submanifold. We classify all submanifolds which at every point realize the equality in the above inequality. As a direct consequence of our Theorem, we give a new characterization 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

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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 space9 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) (19 1 O : S ~ - ~ C ~, ~ ( x l , . . . , X ~ + l ) - l + x n + 2 1 (x l , x l x n + l , " "" 'Xn'XnXn+l) 9

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

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

(1.2)

II 0

(I)(Xl,...

0>0,

( x , , ,xn),0co(l+ xL1) + iXn+ CO+ iSOXn+l '

C~ ..~ 80Xn+ 2 2 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 c p n ( 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 ) , (1.3)

Oo(xl,

Xn+l)=I-i~ 9 '',

0 > 0,

(xl'''''xn)'sOcO(1-+x2+l)--ixn+l~ ' ~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 - y n2 = - 1 } Example4: I f R H n-~ = {y = ( y ~ , . . . , y n ) e R n : y2 + ' ' ' + yn_~ denotes the (n - 1)-dimensional real hyperbolic space, following [2] (cf. [9]), we

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define a one-family of Lagrangian embeddings ~ : $1

•

RHn-1 ~ C H ~ ( - 4 ) ,

fl 9 (0,7r/4],

given by (1.4)

q~(eU,y) = II o (

1 t (cos/~ cos t - isinflsint;y) ) sin fl cos t + i cos fl sin

where H: H12~+l --+ C H n ( - 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 + i g ~ t )

where el = 31 ( 0 , . . . , 0 , 1 , - 1 ) , e~ = ~1 ( 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 CIEn , 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. MAIN THEOREM:

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

manifold. Then

(1.6)

[X~hl2 > ~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* = 0,1 < i , j , k , l =< A~X,Y >.

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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])

V~cJY : J V x Y ,

(2.3) (2.4)

AjxY

=

- J h ( X , Y)

= AjyX.

h(X,Y),JZ

The above formulas immediately imply that < symmetric, i.e.,

(2.5)

> is totally

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

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*,. 99 en* are normal vector fields given by (2.6)

el. : Jel~...,en*

-- flen.

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*

h i j = h k j = hik ,

(2.5)'

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 j*

(2.7)

R m i l p -': C(~ml(~ip - ~mp~il) § E ( h m l h i p

(2.8) (2.9)

Ri*j*kl

j * __

J k* , l < _ i , j , k , l < n , hk~l=hil,j X-"r~i* ~J* ---- C(SjlSik -- 5jkSil) + ff_.~\,~mk,~ml

,hJ*~mp,hJ*),~il

i* j*

-- h m l h m k ),

m

where hik-[l is defined by

k.

(2.10) l

k"

Eh l

;e.§

" l

m

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

( V x h ) ( Y , Z ) = V ~ h ( Y , Z ) - h ( V x Y , Z) - h(Y, V x Z ) ,

(2.10)'

where X , Y , Z are tangent vector fields on M. We note that (Vekh)(ei,ej) = k* totally symmetric, i.e., Combining (2.5)' with (2.8), we know hij,l

hk~l ---- hjl,k i* = h j" lk,i = h l•ki,j,

(2.11)

1 = 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 t h a t at the vector e2 the function ](v) : = < h(v,v), Jv >, < v , v >= 1, restricted to V(A2) attains its maximal value. Let v=coste2+sintei,

0 = 5ij,

0 -~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,

i#jk2.

Thus we have ==0,

i>2,

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

i,j>2,

and that is, we can write (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) (3.57)

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

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Isr. J. Math.

0 Ai(s) = -~s < h(Ex,Ei),JEi > = < (VElh)(E1,Ei),JEi > = #,

By use of that (3.59)

(3.57)

and (3.58), taking the derivative of

(3.56)

i _> 2.

along "/(s) implies

, ( 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* ---- 0, i.e., from which we conclude that # = 0. From (3.11) we know that hij,l 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 PROPOSITION 4.1: Let x: M --+ Nn(4c) be an n-dimensional Lagrangian submanifold; 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, el*,...,en* with el. parallel to H such that the second fundamental form of M in Nn(4c) takes the following form: (4.2)

h(el,el) = ~le.,

h(e~,e~) . . . . .

h(el,ej)=A2ej.,

h(ej,ek)=O,

with (4.3) (~7xh)(Y,Z)=#( JX+<X,Z> and d# vanishes nowhere.

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

2<j~k

JZ),

#r

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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 # J e l

and (4.5)

(Ve~h)(el,el) = #Jei,

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#, A1 -- 2A2

i > 2.

Now choosing X = Y = Z = el, i > 2 in (4.3), we have (4.8)

(V~h)(e~,ei) = 3 # J e ,

i _> 2.

From (2.10)' and (4.2), we get by use of (4.7) ((Ye~ h)(ei, ei) = V ~ h(ei, ei) - 2h(Ve~ ei, ei) = V ~ ( A 2 J e l ) - 2 < el,Ve~ei > h(el,ei) -

(4.9)

2~

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

1>2

=ei(A2)Jel + A1A2# -2A-~ Je~ - 2 < el,Ve~e~ > A2Jei =ei(A2)Jel +

3A2# Jei, A1 - 2A2)

It follows from (4.8) and (4.9) that p-

A2# A1 - 2A2"

i > 2.

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H. LI A N D L. V R A N C K E N

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As # # 0, we have (4.10)

A1 = 3A2.

By use of (4.10), we now can easily check that in this case hikj. _ n n+ 2(Hk.5~ J + Hi" 5kj + H ~9 5ik), which is equivalent to 2(n + 2)

n + 2

H2=n-~(n--1)R-

n

c,

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* = hij,k t" , hij,l

(5.1)

Hk* i* ,~ : H,k.

Thus (3.4) is equivalent to j.

(5.2)

t* -- n n+ 2 (H'ki*hjl + H,k 5il + H~hij), hij'k

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

Define L~jl and its covariant derivative Lijl,k as follows: (5.3)

(5.4)

L~jl = h~ - - - -n- ~ ( H n+

i*

5jt + HJ*hiz + H l* 5ij),

l