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 <_ n; (t) M has parallel second fundamental form, i.e., hiy,~ or
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(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 p n ( 4 ) , given by (1.2); (2.3) in case c = - 4 , x ( M ) is an open portion of one of the Lagrangian submanifolds in C H n ( - 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 I hl 2 =
3n 2 n+2
IV•
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 c p n ( 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. Preliminaries
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)
fTxY=VxY+h(X,Y)
and
fTx~=-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 >.
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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 <_i , j , k , l <_n.
We also have the following formulas: (2.12)
hkf,tP- hk~pI= E h~jR.~i~p+ E h~:Rmjl.+ E h~j*R~*k*Ip, m
m
Rm*i*~p = Rmitp,
(2.13) 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
E H f O, = dH a• + E Ht'Ot'k*'
(2.15)
l
where H k* H ,lk* : - H , kI* "
1
l
~-~i h~/*. Moreover, as a consequence of (2.11), we have that
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 } ,
f(e ) :
0.
then
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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
AJe ~ .
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. LEMMA 3.2: Let x: M --+ N'~(4c) be an n-dimensional Lagrangian submanifold.
Then 3n2 i V l ~ l 2 '
I~h12 > n + 2
(3.3)
where I~'hl 2 = ~i,j,k,t(hij,t) k* 2 , ]V• holds in (3.3) if and only if (3.4)
= ~k,i(H,~*) ~. Moreover, the equality
hij, _ re +n 2 (H k* 5jl + H,jk* 5~1 + H,tk* 5~j),
1 _< i, j, k, l _< n.
Proof: We construct a tensor W by (3.5)
W~; := hij,lk* _ ~-~"(H+ k*i'(~Jntz
k*
k*
+ H,j 5it + H,t 5ij).
It is easy to check that 3n2 (3.6)
0 < IWl 2 := ~
(w~)2
= iVhl 2 _ _r ~_ 2 1 v
•
~
2
HI,
i,j,k,l
where IVhl 2 -~ ~i,j,k,l(hij,l) k* 2 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).
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LEMMA 3.3: Let x: M ~ N n ( 4c) be an n-dimensional Lagrangian submanifold. If (3.4) holds, then H,ji* : ,~(~ij
(3.7)
for some function A on M , i.e., the vector field - J H = ~ k Hk* ek is a conformal vector field. Proof'. (3.8) (3.9)
From (3.4), we have for all i, j, k, l hk;, l _
n 2 ( H f h j L + Hkj.5, l + H~*hij),
re+
l*
'
re
l*
'
l*
4"
hij'k - n + 2 (H,i 5jk + H 'j ~ik "b H ' k 5ij).
k* From (2.11), we have h~,k = hij,l , therefore (3.8) and (3.9) imply
(3.10) H i,k* 5jlq-H,jk* 5 i l + H l,k* 5ij -- Hl;hjk,
+Hl;hik 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 notation of [18], we note that 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. If (3.4) holds, then (3.11) hk~l = #(hkihjl + 5kjhit + 5klhij),
#=n+2
1
1 < i , j , k , 1 < re,
p
(3.12)
e~(#) = 0,
(3.13)
- el(p) + (2Al - A1)(c+ AIA1 - A~) = 0,
-
/ : 2,...,n, 1 = 2,...,n.
If Ajepe~ ~ 0 for s o m e p , l, 2 <_p ~ l ~ n, we have (3.14)
el(#) = 0,
in which case (3.12) and (3.13) imply that # is a constant. Proof: The first statement (3.11) follows immdiately from (3.7) and (3.4). Taking the covariant derivative of (3.11) implies that
(3.15)
k*
hij,l p : ep(#)(hkihjl + ~kjhil -b ~kl~ij).
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Exchanging I with p in (3.15), we have
h~p l = el (#)(hk~hj~ + 5kjhi~ + 5k~5ij).
(3.16)
Putting (2.7), (2.13), (2.9) into (2.12), we have k*
k*
k*
m*
m
m
k* =e(~13 ~
(3.17)
+
~.k* ~
-
m
~pj~~"
~ ~)
-
+ e(h~j~kp P* (~kl) T ff_~ ~ ,Ojm,Olm,~ip ]~k* hn* l~n* -- ~ ~ ,~jm,Omp,~il ~k* ]~n* ~n* * -- hij m,n m,n E
~k* ~n* ~n*
m,n
m,n m* n* n*
n* n* + E hij hm~hkp - E hijm* hmp hkl" m,n
m,n
Substituting (3.15) and (3.16) into (3.17), we find that
k* -----c(hlj (~ip "[- h~( (~jp --
(3.18)
k*
+ c(h~}hkp - h~; 5kt) + ~ ~'176176~ * ~ "
_ ~hk*
m,n k*
n*
n*
Z...,
hn* ~n*
m,n
k*
n*
r~*
+ E himhlmhjp - E himhmphj l m~n m,n m* n* n*
n*
m~n
n*
m~n
Now we take an orthonormal frame as in Lemma 3.1. Then we have hij1" )~iSij,1 <_ i,j <_ n. Choosing i = j = p = 1,l r 1 in (3.18), we get that (3.19)
e~(~)hkl -- 3 e l ( ~ ) h k ~
=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,
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which is equivalent to
(Ap-At)(AiAj%el-2AjelAj%el)=O, p # l ,
(3.21)'
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 - A2 < 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:
A2 =
(3.24)
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 + 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
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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 eigenvalue A1/2).
If Aje~ea # 0 for some a , a , from L e m m a 3.4, we conclude
el(#) = 0. Combining with (3.20), we get # is constant, which is a contradiction 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 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
3
We have the function
g(t) := f(coste2 + sintei). It is easy to check t h a t g'(0) = 0 is equivalent to (3.29)
= < A g e 2 e 2 , e i > = O ,
3
From (3.29), we can therefore assume that
Aj~2e2 = A2el + k2e2.
(3.30)
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)
el(,)
-- (2/~ 2 - / ~ 1 ) ( c ~- )tl/~ 2 - ,~22).
Since we assumed t h a t el(#) # 0, (3.32) implies that c + ~1~2 - A2 # 0. Thus it follows from (3.31) t h a t (3.33)
k2 = 0.
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However, (3.30) and (3.33) then imply that
f(e2) = < h(e2, e2), Je2 > = 0.
(3.34)
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,
Proos (i) I f i = a , j = b , k = c o r i = a , j = / 3 ,
2 _< i , j , k <_n.
k=%weget
(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
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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. Under the same assumptions as in L e m m a 3.5, let
LEMMA 3.9:
A2 . . . . .
A j e l e i = Aiei,
Ar+l # Ar+2 . . . . .
An;
then
(3.38)
A1 -
3Aa
2 < a < r + 1,
iS constant,
and
(3.39) Prod:
(3.40)
Aa-An
is constant,
r+2
2 < a < r +1,
From (2.11), assumption condition (3.4) is equivalent to k* _
hij'l
n
n+2
j*
k*
(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" hll,t - 3 ~ - - ~ H , l = 0,
(3.41)
l
Choosing k = 1,i = j _> 2 in (3.40), we have (3.42)
h 1" ~i,z-~ n n+
H 1,l. = 0 ,
2
l
Thus we have from (3.41) and (3.42) (3.43)
h 1. 115 - 3 h ~ l = 0 ,
l
1" From definition (2.10) and (3.37), we have h~[,l = el(A1), hii,l = ez(Ai), i = 2 , . . . , n, 1 < I < n. Thus we have from (3.43)
(3.44)
et(A, - 3Ai) = 0,
i = 2,...,n,
l
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).
|
2
r+l<~
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: (3.46)
If H is not constant, it follows from the previous lemma t h a t is not a constant, i.e., dAa # O.
Aa
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).
B y (3.39), we can introduce a constant K by (3.48)
K := Am - Aa = constant.
P u t t i n g (3.48) into (3.47), we get (3.49)
K[2c - 6A2a - 6KAa - 2 K 2 + 6AIAa + 3KA1 - A12]= 0.
(3.48) and assumption condition (3.25) imply K # 0, thus we have (3.50)
2c - 6A~ - 6KAa - 2K 2 + 6A1Aa + 3KA1 - AT = 0.
Differentiating (3.50) and using (3.38), we have (3.51)
(2Aa+K)dAa=0,
a=2,...,r+l,
which is a contradiction with (3.46). Thus H =constant. Since H is constant, we have from L e m m a 3.1 and L e m m a 3.8 t h a t
= 1E
h(ek,ek)
nl ( E hiki k * k J e i ) "] l J e l '
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 (3.53)
H,11" = 0.
Combining (3.53) with (3.7), we have A = 0, i.e., H k* = 0. We conclude now from (3.4) t h a t h~,k = 0, i.e., the second fundamental form h of M is parallel. |
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237
PROPOSITION 3.1: Let x: M --4 N'~(4c) be an n-dimensional Lagrangian submanifold. If (3.11) holds with # =constant, then # = 0 and M is of parallel second fundamental form. Proof: By the definition of #, (3.11) and (2.10)', we have (3.54)
< (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 { E l , . . . , 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) < Ei,Ej > = 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#
238 and (3.58) o
H. LI AND L. VRANCKEN
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<_n,
JY+<X,Y>
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.
240
H. LI A N D L. V R A N C K E N
Isr. J. M a t h .
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 <_i,j,l <_n,
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
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THEOREM 5.1: Let x: M -4 Nn(4c) be an n-dimensional Lagrangian submanifold. If
Lijl,k =O,
(5.5)
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 p n ( 4 ) , given by (1.2). (2.3) /n case c = - 4 , x ( M ) is an open portion of one of the Lagrangian submanifolds in C H ~ ( - 4 ) , given by (1.3), (1.4) and (1.5).
ACKNOWLEDGEMENT: T h e authors express their thanks to Udo Simon for his help. References
[1] V. Borrelli, B. Y. Chen and J. M. Morvan, Une caractdrization gdomdtrique de la sphere de Whitney, Comptes Rendus de l'Acad~mie des Sciences, Paris, S~rie I, Math~matique 321 (1995), 1485-1490. [2] I. Castro, C. R. Montealegre and F. Urbano, Closed conformal vector fields and Lagrangian submanifolds in complex space forms, Pacific Journal of Mathematics 199 (2001), 269 302. [3] I. Castro and F. Urbano, Lagrangian surfaces in the complex Euclidean plane with conformal Maslov form, The T6hoku Mathematical Journal 45 (1993), 565-582. [4] I. Castro and F. Urbano, Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes, Annals of Global Analysis and Geometry 13 (1995), 59-67. [5] B. Y. Chen, Jacobi's elliptic functions and Lagrangian immersions, Proceedings of the Royal Society of Edinburgh. Section A 126 (1996), 687-704. [6] B. Y. Chen, Interaction of Legendre curves and Lagrangian submanifolds, Israel Journal of Mathematics 99 (1997), 69 108.
242
H. LI AND L. VRANCKEN
Isr. J. Math.
[7] B. Y. Chen, Complex extensors and Lagrangian submanifolds in complex Euclidean spaces, The TShoku Mathematical Journal 49 (1997), 277-297. [8] B. Y. Chen and K. Ogiue, On totally real submanifolds, Transactions of the American Mathematical Society 193 (1974), 257-266. [9] B. Y. Chen and L. Vrancken, Lagrangian submanifolds satisfying a basic equality, Proceedings of the Cambridge Philosophical Society 120 (1996), 291-307. [10] R. Harvey and H. B. Lawson, Calibrated geometries, Aeta Mathematica 148 (1982), 47-157. [11] G. Huisken, Flow by mean curvature of convex surfaces into spheres, Journal of Differential Geometry 20 (1984), 237-266. [12] H. Li, Willmore surfaces in S n, Annals of Global Analysis and Geometry 21 (2002), 203-213. [13] H. Li, Willmore submanifolds in a sphere, Mathematical Research Letters 9 (2002), 771-790. [14] S. Montiel and 17. Urbano, Isotropic totally real submanifolds, Mathematische Zeitschrift 199 (1988), 55-60. [15] H. Naitoh, Totally real parallel submanifolds, Tokyo Journal of Mathematics 4 (1981), 279-306. [16] H. Naitoh, Parallel submanifolds of complex space forms I, Nagoya Mathematical Journal 90 (1983), 85-117; II, ibid 91 (1983), 119-149. [17] H. Naitoh and M. Takeuchi, Totally real submanifolds and symmetric bounded domain, Osaka Journal of Mathematics 19 {1982), 717-731. [18] A. Ros and F. Urbano, Lagrangian submanifolds of C n with conformal Maslov form and the Whitney sphere, Journal of the Mathematical Society of Japan 50 (1998), 203-226.
