A variational problem for submanifolds in a sphereГ

Monatsh. Math. 152, 295–302 (2007) DOI 10.1007/s00605-007-0476-2 Printed in The Netherlands

A variational problem for submanifolds in a sphere
By

Zhen Guo1 and Haizhong Li2 1

Yunnan Normal University, Kunming, P.R. China 2 Tsinghua University, Beijing, P.R. China Communicated by D. V. Alekseevsky

Received September 3, 2006; accepted in revised form November 21, 2006 Published online June 11, 2007 # Springer-Verlag 2007 Abstract. Let x: M ! Snþp be an n-dimensional submanifold in an ðn þ pÞ-dimensional unit a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the sphere Snþp , M is called Ð P n Willmore functional M ðS  nH 2 Þ2 dv, where S ¼
;i; j ðh
ij Þ2 is the square of the length of the second fundamental form, H is the mean curvature of M. In [11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in Snþp . In this paper, we discover that Ð a similar integral inequality of Simons’ type still holds for the critical submanifolds of the functional M ðS  nH 2 Þdv. Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation. 2000 Mathematics Subject Classification: 53C42, 53A10 Key words: Euler-Lagrangian equation, integral inequality, Clifford torus, Veronese surface

1. Introduction For brevity, we use the same notations as [11] in this paper. Let M be an n-dimensional compact submanifold of an ðn þ pÞ-dimensional unit sphere space Snþp . If h
ij denotes the second fundamental form of M, S denotes the square of the length of the second fundamental form, H denotes the mean curvature vector and H denotes the mean curvature of M, then we have X X 1X
S¼ ðh
ij Þ2 ; H¼ H
e
; H
¼ h ; H ¼ jHj; n k kk

;i; j where e
(n þ 1 4
4 n þ p) are orthonormal normal vector fields of M in Snþp . We define the following non-negative function on M, 2 ¼ S  nH 2 ;

ð1:1Þ


Z. Guo is supported by the grant No. 10261010 of NSFC and H. Li is supported by the grant No. 10531090 of the NSFC and by Doctoral Program Foundation of the Ministry of Education of China (2006).

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which vanishes exactly at the umbilic points of M. The Willmore functional (see [11], [5], [16]) is ð ð n WðxÞ ¼ n dv ¼ ðS  nH 2 Þ2 dv; ð1:2Þ M

M

the Euler-Lagrange equation (i.e. Willmore equation) can be found in (1.2) of [11]. In this paper, we consider the following non-negative functional ð ð 2 FðxÞ ¼  dv ¼ ðS  nH 2 Þdv; ð1:3Þ M

M

which vanishes if and only if M is a totally umbilical submanifold, so the function FðxÞ measures how derivation xðMÞ is from totally umbilical submanifold. Remark 1.1. When n ¼ 2, FðxÞ reduces to the well-known Willmore functional WðxÞ, and its critical points are called Willmore surfaces. The Willmore surfaces in a sphere were studied by Thomsen [20], Willmore [22], Bryant [3], Pinkall [17], Weiner [21], Montiel [15], Li [7], Li and Simon [12], Li and Vrancken [13] and many others (also see Blaschke [2]). In this paper, we first calculate the Euler-Lagrangian equation of FðxÞ given by (1.3). Theorem 1.1. Let x: M ! Snþp be an n-dimensional submanifold in an ðn þ pÞ-dimensional unit sphere Snþp . Then M is a extremal submanifold of FðxÞ if and only if for n þ 1 4
4 n þ p X X n ðn  1Þ
? H
þ h
ij hik hkj  H  hij h
ij  2 H
¼ 0; ð1:4Þ 2 ;i; j;k ;i; j where
? H
¼

P

i


H;ii ( for notations here, see [11]).

We call x: M ! Snþp an extremal submanifold if it satisfies Euler-Lagrange equation (1.4). Remark 1.2. When n ¼ 2, Theorem 1.1 was proved by Weiner in [21]. In this case (1.4) reduces to the following well-known equation of Willmore surfaces (see [21] or [7]) X
? H
þ h
ij hij H   2H 2 H
¼ 0; 3 4
4 2 þ p: ð1:5Þ ;i; j

Remark 1.3. It is remarkable that when n 5 3, the Euler-Lagrange equation (1.4) of the functional FðxÞ is much simpler that the Willmore equation (1.2) of [11]. In order to state our main result, we recall the following important examples Example 1 (see [4] or [6]). The Clifford torus
rffiffiffi
rffiffiffi 1 1 m m Cm;m ¼ S S ; 2 2

n ¼ 2m;

ð1:6Þ

A variational problem for submanifolds in a sphere

297

is an extremal hypersurface in Snþ1 . In fact, the principal curvatures k1 ; . . . ; kn of Cm;m are k1 ¼    ¼ km ¼ 1; kmþ1 ¼    ¼ kn ¼ 1; n ¼ 2m: ð1:7Þ We have from ð1:7Þ H ¼ 0;

S ¼ n;

X

ki3 ¼ 0:

ð1:8Þ

i

Thus we easily check that ð1:4Þ holds, i.e., Cm;m is an extremal hypersurface. In particular, we note that 2 of Cm;m satisfy 2 ¼ n:

ð1:9Þ

Example 2 (see [4] or [7], [11]). The Veronese surface satisfies (1.5) and 2 ¼ 43. Example 3. If x: M ! Snþp is a minimal surface or an n-dimensional ðn 5 3Þ Einstein and minimal submanifold, then it must be an extremal submanifold. It can be checked directly that in this case (1.4) is satisfied by use of Gauss equation and minimal condition H
¼ 0. In [10], [11], the second author proved the following integral inequality of Simons’ type: Theorem 1.2 ([10], [11]). Let M be an n-dimensional ðn 5 2Þ compact Willmore submanifold in ðn þ pÞ-dimensional unit sphere Snþp . Then we have
 ð n n 2   dv 4 0: ð1:10Þ  2  1=p M In particular, if 0 4 2 4

n ; 2  1=p

ð1:11Þ

n then either 2  0 and M is totally umbilical, or 2  21=p . In the latter case, either p ¼ 1 and M is a Willmore torus Wm;nm , or n ¼ 2, p ¼ 2 and M is the Veronese surface.

In this paper we discover that a similar integral inequality of Simons’ type still holds for compact extremal submanifolds in Snþp . Theorem 1.3. Let M be an n-dimensional (n 5 2) compact extremal submanifold in ðn þ pÞ-dimensional unit sphere Snþp . Then we have
 ð n 2 2   dv 4 0:  ð1:12Þ 2  1=p M In particular, if 0 4 2 4

n ; 2  1=p

ð1:13Þ

n then either 2  0 and M is totally umbilic, or 2  21=p . In the latter case, either p ¼ 1, n ¼ 2m and M is a Clifford torus Cm;m defined by ð1:6Þ; or n ¼ 2, p ¼ 2 and M is the Veronese surface.

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Remark 1.4. When n ¼ 2, Theorem 1.3 was proved by the second author in [7] (also see Li and Simon [12]). 2. Proof of Theorem 1.1 We use the same notations as in [11]. Let x0 : M ! Snþp be an n-dimensional compact submanifold. Now we calculate the first variation of the functional Fðx0 Þ Let x: M  R ! Snþp be a smooth variation of x0 such that xð; tÞ ¼ x0 on the boundary. Along x: M  R ! Snþp , we choose a local orthonormal basis feA g for TSnþp with dual basis f!A g, such that fei ð; tÞg forms a local orthonormal basis for xt : M  ftg ! Snþp. Since T
ðM  RÞ ¼ T
M  T
R, the pullback of f!A g and f!AB g on Snþp through x: M  R ! Snþp have the decomposition x
! ¼  þ a dt; ð2:1Þ x
! ¼ a dt;


x
!ij ¼ ij þ aij dt;




i

i

x
!i
¼ i
þ ai
dt;

i

x
!




¼ 
 þ a
 dt;

ð2:2Þ

where fai ; a
; aij ; ai
; a
 g are local functions on M  R with aij ¼ aji , a
 ¼ a
and  X X d  V ¼  xt ¼ ai dx0 ðei Þ þ a
e
; ð2:3Þ dt t¼0
i is the variation vector field of xt : M ! Snþp . We note that the one forms fi ; ij ; i
; 
 g are defined on M  ftg, for t ¼ 0, they reduce to the forms with the same notation on M. We denote by dM the differential operator on T
M, then we have d ¼ dM þ dt @t@ on T
ðM  RÞ. Lemma 2.1. Under the above notations, we have X @i X ¼ ðai; j þ aij Þj  h
ij a
j ; @t j j;
X ai
¼ a
;i þ h
ij aj ;

ð2:4Þ ð2:5Þ

j


 X X @i
X ¼ aik h
jk  a
hij þ a
ij j ; ai
; j þ @t j k 

ð2:6Þ

where h
ij and the covariant derivatives ai; j , a
;i and ai
; j are defined on M  ftg by X h
ij j ; ð2:7Þ i
¼ X

j

ai; j j ¼ dM ai þ

j

X

j

aj ji ;

ð2:8Þ

a 
;

ð2:9Þ

j

a
;i i ¼ dM a
þ

i

X

X

ai
; j j ¼ dM ai
þ

X j

X 

aj
ji þ

X 

ai 
:

ð2:10Þ

A variational problem for submanifolds in a sphere

299

Proof. These are direct calculations. In fact, substituting (2.1) and (2.2) into the following equations, respectively, 
X X !ij ^ !j þ !i
^ !
; dðx
!i Þ ¼ x
ðd!i Þ ¼ x



j

dðx
!
Þ ¼ x
ðd!
Þ ¼ x



X

!
j ^ !j þ

X

j

dðx
!i
Þ ¼ x
ðd!i
Þ ¼ x



X

 !
 ^ ! ;



!ij ^ !j
þ

j

X

 !i ^ !
 !i ^ !
;



and comparing the terms in T
M ^ dt for each equation on the both sides, we can get (2.4), (2.5) and (2.6), respectively. Lemma 2.2.  X X
X @h
ij ¼ a
;ij þ aik h
kj þ ajk h
ki þ h
ijk ak þ a
 hij þ ij a
þ h
ik hkj a : @t k  k; ð2:11Þ Remark 2.1. When p ¼ 1, Lemma 2.2 was proved by Barbosa and Colares [1] (see Lemma 6.1 of [1]). We also note that the sign of the second fundamental form here is different from theirs. Proof. Differentiating (2.7) with respect to t and using (2.4) and (2.6), we get @h
ij @t

¼ ai
; j þ

X k

aik h
jk 

X

a
hij þ a
ij 

X X ðh
ik ak; j þ h
ik akj Þ þ h
ik hkj a : k



k;

Covariant differentiating (2.5) over M  ftg and using the Codazzi equation for xt : M ! Snþp , we get X ðak; j h
ik þ ak h
ikj Þ ai
; j ¼ a
;ij þ k X ðak; j h
ik þ ak h
ijk Þ: ¼ a
;ij þ k

Combining the above two equations, we prove Lemma 2.2. P Set i ¼ j in (2.11) and making summation over i with using i;k aik h
ki ¼ 0, we get X X @H
1 ? 1X
 ¼
aa þ H;k
ak þ a
 H  þ h h a þ a
: ð2:12Þ n n i;k; ik ki @t k  From (2.11) and the fact that X X ðh
ij Þ2 ; a
 hij h
ij ¼ 0; S¼ i; j;


i; j;
;

X i; j;k;


ajk h
ki h
ij ¼ 0;

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we obtain X 1 @S X
1X ¼ hij a
;ij þ S;k ak þ nH
a
þ h
ij h
ik hkj a : 2 @t i; j;
2 k i; j;k;
; From (2.12) and

P

ð2:13Þ

a
 H
H  ¼ 0, we obtain


;

X X n @H 2 X
? nX 2 ¼ H
a
þ ðH Þ;k ak þ H
h
ij hij a þ n H
a
: ð2:14Þ 2 @t 2 k

i; j;
; For xt : M ! Snþp, we consider the functional ð ð 2 Fðxt Þ ¼  dv ¼ ðS  nH 2 Þ1 ^    ^ n : M

ð2:15Þ

M

From (2.4), we have X @ @i ð1 ^    ^ n Þ ¼ 1 ^    ^ ^    ^ n @t @t i X ¼ ðai;i þ aii  h
ii a
Þ1 ^    ^ n i

¼


X

ai;i  n

X






H a
1 ^    ^ n :

ð2:16Þ




i

Differentiating (2.15) with respect to t, we get by use of (2.13), (2.14) and (2.16)  ð  X X X X @Fðxt Þ ¼ 2 h
ij a
;ij  2 H

? a
þ ð2 Þ;k ak þ 2 ak;k @t M
i; j;
k k   X X   X n þ2 hij hik h
kj  H  hij h
ij  2 H
a
dv ð2:17Þ 2
i; j;k; i; j; We note that

X X X

2 ak ;k ; ð2 Þ;k ak þ 2 ak;k ¼ k

k

and M is compact (without boundary), also noting X X h
ijj ¼ nH;i
; h
ijji ¼ n
? H
; j

ð2:18Þ

k

ð2:19Þ

i; j

it follows from (2.17), (2.18) and Green’s formula that  ð X X X @Fðxt Þ n 2
 
 
?
¼2 hij hik hkj  H hij hij   H þ ðn  1Þ
H a
dv @t 2 M
i; j;k; i; j; ð2:20Þ From (2.3) and (2.20) with restriction to t ¼ 0, we have proved Theorem 1.1.

A variational problem for submanifolds in a sphere

301

3. The Lemmas and Proof of Theorem 1.3 Define tensors

~
 ¼

X

~h
¼ h
 H
ij ; ij ij ~h
~h ; ij ij


 ¼

i; j

ð3:1Þ X

h
ij hij :

ð3:2Þ

i; j

By use of Theorem 1.1, (3.1) and (3.2), we can get Lemma 3.1. Let M be an n-dimensional submanifold in an ðn þ pÞ-dimensional unit sphere Snþp . Then M is an extremal submanifold if and only if it satisfies for n þ 1 4
4 n þ p X
  X n ~ H  ~
  H
2 þ H
2 : ð3:3Þ hij ~ hik ~ hkj ¼ ðn  1Þ
? H
 2 ;i; j;k  Lemma 3.2 (see Lemma 4.5 of [11]). Let x: M ! Snþp be an n-dimensional submanifold in Snþp . Then X X 1 2 


 5 jrhj2  n2 jr? Hj2 þ ðh
ij h
kki Þj þ n H  ~hmj ~hij ~him 2
;i; j;k
;;i; j;m 
1 1 4 
ðnH 2 Þ: þ n2 þ nH 2 2  2  ð3:4Þ p 2 From (3.3), we have X X X 

~ j2  2 H ~
? H   H   n H
H  ~
  njH hmj ~ hij ~ him ¼ nðn  1Þ n
;;i; j;k


; 2

n ~2 2 jH j  : ð3:5Þ 2 Integrating (3.4) over M and using Stokes’ formula, we have by use of (3.5), ð  X X jrhj2  n2 jr? Hj2  nðn  1Þ 05 H

? H
 n H
H  ~
 þ n2 M  
n2 2 2 1 4  dv þ H   2 p 2  

 ð  1 2 nðn  2Þ 2 2  þ H  dv 5 nð2 H 2  H
H  ~
 Þ þ 2 n  2  p 2 M   
ð 1 2  dv; 2 n  2  ð3:6Þ 5 p M þ

where we used Lemma 4.2 of [11] and ~
 ¼ ~

 (see (4.9) of [11]) X X X X H
H  ~
 ¼ ðH
Þ2 ~
4 ðH
Þ2  ~ ¼ H 2 2 :
;









Thus we reach the integral inequality ð1:12Þ in Theorem 1.3.

ð3:7Þ

302

Z. Guo and H. Li: A variational problem for submanifolds in a sphere

If ð1:12Þ that either 2  0, or

holds, then we conclude from ð5:4Þ 1 2   n= 2  p . In the first case, we know that S  nH , i.e. M is totally umbilic; in the latter case, i.e., 
X
2 1 2 ~ ; ð3:8Þ  ¼ ðhij Þ  n= 2  p
;i; j 2

ð3:6Þ becomes an equality, we conclude that either H ¼ 0 if n 5 3, or n ¼ 2. If n 5 3 and H ¼ 0, from Main Theorem of Chern, Do Carmo and Kobayashi [4] we get p ¼ 1 and
rffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffi m nm m nm M¼S S ; ð3:9Þ n n combining (3.9) with (1.4), we conclude that n ¼ 2m and M ¼ Cm;m , thus we prove Theorem 1.3. If n ¼ 2, our Theorem 1.3 comes from Theorem 3 of [7] (also see [12]). We complete the proof of Theorem 1.3. References [1] Barbosa JLM, Colares AG (1997) Stability of hypersurfaces with constant r-mean curvature. Ann Global Anal Geom 15: 277–297 [2] Blaschke W (1929) Vorlesungen € uber Differential Geometrie-III. Berlin: Springer [3] Bryant RL (1984) A duality theorem for Willmore surfaces. J Differential Geom 20: 23–53 [4] Chern SS, Do Carmo M, Kobayashi S (1970) Minimal submanifolds of a sphere with second fundamental form of constant length. In: Brower F (ed) Functional Analysis and Related Fields, pp 59–75. Berlin: Springer [5] Guo Z, Li H, Wang CP (2001) The second variational formula for Willmore submanifolds in Sn . Results Math 40: 205–225 [6] Lawson HB (1969) Local rigidity theorems for minimal hypersurfaces. Ann Math 89: 187–197 [7] Li H (2002) Willmore surfaces in Sn . Ann Global Anal Geom 21: 203–213 [8] Li H (1996) Hypersurfaces with constant scalar curvature in space forms. Math Ann 305: 665–672 [9] Li H (1997) Global rigidity theorems of hypersurfaces. Ark Mat 35: 327–351 [10] Li H (2001) Willmore hypersurfaces in a sphere. Asian J Math 5: 365–378 [11] Li H (2002) Willmore submanifolds in a sphere. Math Research Letters 9: 771–790 [12] Li H, Udo S (2003) Quantization of curvature for compact surfaces in Sn . Math Z 245: 201–216 [13] Li H, Vrancken L (2003) New examples of Willmore surfaces in Sn . Ann Global Anal Geom 23: 205–225 [14] Li P, Yau S-T (1982) A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent Math 69: 269–291 [15] Montiel S (2000) Willmore two-spheres in the four-sphere. Trans Amer Math Soc 352: 4469– 4486 [16] Pedit F, Willmore TJ (1988) Conformal geometry. Atti Sem Mat Fis Univ Modena 36: 237–245 [17] Pinkall U (1985) Hopf tori in S3 . Invent Math 81: 379–386 [18] Reilly RC (1973) Variational properties of functions of the mean curvature for hypersurfaces in space forms. J Differential Geom 8: 465–477 [19] Simons J (1968) Minimal varieties in riemannian manifolds. Ann Math 88: 62–105 € ber Konforme Geometrie. I: Grundlagen der konformen Fl€achentheorie. [20] Thomsen G (1923) U Abh Math Sem Hamburg 3: 31–56 [21] Weiner J (1978) On a problem of Chen, Willmore, et al. Indiana Univ Math J 27: 19–35 [22] Willmore TJ (1965) Notes on embedded surfaces. Ann Stiint Univ Al I Cuza, Iasi, Sect I a Mat (NS) 11B: 493–496 Authors’ addresses: Zhen Guo, Faculty of Mathematical Sciences, Yunnan Normal University, 650092 Kunming, People’s Republic of China, e-mail: [email protected]; Haizhong Li, Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, People’s Republic of China, e-mail: [email protected]