Vanishing theorems on hypersurfaces in Riemannian manifolds

Nonlinear Analysis 75 (2012) 5039–5043

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Vanishing theorems on hypersurfaces in Riemannian manifolds✩ Peng Zhu ∗ School of Mathematical Sciences, Yangzhou University, Yangzhou, Jiangsu 225002, PR China

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Article history: Received 31 October 2011 Accepted 10 April 2012 Communicated by S. Carl MSC: 53C21 54C42

abstract We study a complete noncompact stable minimal hypersurface M and a strongly stable hypersurface M with constant mean curvature in a 5-dimensional Riemannian manifold 5 ≤ N. If N is a compact simply connected manifold with bounded sectional curvature 17

K¯ ≤ 1, then there is no nontrivial L2 harmonic form on M. This is a generalized version of Tanno’s result on a stable minimal hypersurface in R5 and Zhu’s result on a stable minimal hypersurface in S5 . © 2012 Elsevier Ltd. All rights reserved.

Keywords: Harmonic form Constant mean curvature Stable hypersurface

1. Introduction Each complete minimal graph in Rn+1 (n ≤ 7) is a hypersurface. If n ≥ 8, then it is false. Do Carmo and Peng [1] showed that complete orientable stable minimal surfaces in R3 are planes. At the same time, Fischer-Colbrie and Schoen [2] independently showed that a complete stable minimal hypersurface M in a complete 3-dimensional manifold N with nonnegative scalar curvature must be either conformally a plane or conformally a cylinder R × S1 . For the special case, they also prove that M must be a plane if N = R3 . Palmer [3] proved that there is no non-trivial L2 harmonic 1-form on a complete noncompact orientable stable minimal hypersurface in Rn+1 . It implies that there exists some topological obstruction for the stability of M. Tanno [4] showed that there is no nontrivial L2 harmonic 1-form on a complete noncompact orientable minimal hypersurface in N n+1 with non-negative bi-Ricci curvature. Cheng [5] confirmed this result to hold for a complete noncompact orientable strongly stable hypersurface M with constant mean curvature H in an ambient manifold N n+1 with n(n−5) bi-Ricci curvature having a low bound 4 H 2 along M. Tanno [4] proved that if M is a complete orientable stable minimal 5 hypersurface in R then there exist no non-trivial L2 harmonic p-forms on M (0 ≤ p ≤ 4). The author [6] proved that a complete noncompact orientable stable minimal hypersurface in S5 admits no nontrivial L2 harmonic forms and also obtained that a complete noncompact strongly stable hypersurface with constant mean curvature in R5 or S5 admits no nontrivial L2 harmonic forms. In this paper, we consider L2 harmonic forms on a (strongly) stable hypersurface M 4 in a 5-dimensional manifold N 5 whose sectional curvature is bounded. First, we fix some notations so as to state the main result. Suppose that N 5 is a 5-dimensional Riemannian manifold and x : M 4 → N 5 is an isometric immersion of a 4-dimensional orientable manifold M with constant mean curvature H. We denote R¯ , Ric, K¯ , R, Ric and K by the curvature tensor, the Ricci curvature, the sectional curvature of N and the curvature tensor, the Ricci curvature, the sectional curvature of M, respectively. ∇ denotes the LeviCivita connection of M. γ is the unit normal vector field of M. |A| is the normal of the second fundamental form A = (hij ) of x.

✩ This work was partially supported by NSF Grants (China) 11101352, and 11071208.

∗

Tel.: +86 514 87975509; fax: +86 514 87975509. E-mail address: [email protected].

0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.04.019

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P. Zhu / Nonlinear Analysis 75 (2012) 5039–5043

In the minimal hypersurface case, the immersion x is called stable if I ( h) =



|∇ h|2 − (Ric(γ , γ ) + |A|2 )h2 dv ≥ 0,

(1.1)

M

for all compactly supported piecewise smooth functions h on M, where ∇ h is the gradient of h and dv is the volume form. In the case of nonzero constant mean curvature, the immersion x is called strongly stable if (1.1) holds for all compactly supported piecewise smooth functions h on M. The Hodge operator ∗ : ∧p (M ) → ∧4−p (M ) is defined by

∗ei1 ∧ · · · ∧ eip = sgnσ (i1 , i2 , i3 , i4 )eip+1 ∧ · · · ∧ ei4 , where σ (i1 , i2 , i3 , i4 ) denotes a permutation of the set (i1 , i2 , i3 , i4 ) and sgnσ is the sign of σ . The operator d∗ : ∧p (M ) → ∧p−1 (M ) is given by d∗ ω = − ∗ d ∗ ω. The Laplacian operator is defined by

1ω = −dd∗ ω − d∗ dω. A p-form ω is called L2 harmonic if 1ω = 0 and



ω ∧ ∗ω < +∞. M

Denote by H p (L2 (M )) the space of all L2 harmonic p-forms. We obtain that Tanno’s result still holds when the ambient manifold is a 5-dimensional manifold with bounded sectional curvature. More precisely, we obtain the following result. Theorem 1.1. Let M 4 be a complete noncompact orientable stable minimal hypersurface or a complete noncompact orientable strongly stable hypersurface with constant mean curvature in a Riemannian manifold N 5 . If N 5 is a compact simple connected 5 manifold whose sectional curvature K¯ satisfies 17 ≤ K¯ ≤ 1, then H p (L2 (M )) = {0}, for 0 ≤ p ≤ 4. Remark 1.2. From the viewpoint of topology, by pinched theorem, N 5 is only homeomorphism to S 5 in Theorem 1.1. But, N 5 may have different differential structures. 2. Proof of main results We initially introduce two algebraic results which will be used later. Lemma 2.1 ([7]). Let (V m , g ) be an m-dimensional vector space with the metric g. Let Ω ∈ ∧2 V ∗ . There exists an orthonormal basis {e1 , . . . , e2n , . . . , em } (its dual is {e1 , . . . , e2n , . . . , em }) such that

Ω=

n 

αi e2i−1 ∧ e2i ,

i=1

where 2n ≤ m. Lemma 2.2 ([6]). Suppose that A is a symmetric 4 × 4 matrix and B is an antisymmetric 4 × 4 matrix. Then, we have the following relation:

−2trA · trAB2 + 2trA2 B2 + 2tr(AB)2 + |A|2 |B|2 ≥ 0. Let (M , g ) be a 4-dimensional Riemannian manifold. Let {e1 , e2 , e3 , e4 } be locally defined orthogonal frame fields of tangent bundle TM. We denote {e1 , e2 , e3 , e4 } by the dual coframe fields. Suppose ω = ai1 i2 ei2 ∧ ei1 = aI ωI , and θ = bi1 i2 ei2 ∧ ei1 = bI ωI , where the summation is being performed over the multi-index I = (i1 , i2 ), ai1 i2 = −ai2 il and bi1 i2 = −bi2 i1 . Set

⟨ω, θ⟩ =



aI b I

I

and

|∇ω|2 =

4 

|∇ei ω|2 .

i =1

It is known that the following equality always holds for each 2-form ω [8]:

1|ω|2 = 2⟨1ω, ω⟩ + 2|∇ω|2 + 2⟨E (ω), ω⟩,

(2.1)

P. Zhu / Nonlinear Analysis 75 (2012) 5039–5043

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where

⟨E (ω), ω⟩ = 2Ricij ail ajl + 2Rijkl ali akj .

(2.2)

On the other hand, a direct computation shows that

1|ω|2 = 2|ω|1|ω| + 2|∇|ω| |2 .

(2.3)

Thus, (2.1) and (2.3) imply that

|ω|1|ω| = ⟨1ω, ω⟩ + (|∇ω|2 − |∇|ω| |2 ) + ⟨E (ω), ω⟩.

(2.4)

We can apply Lemmas 2.1 and 2.2 to hypersurfaces and obtain the following proposition. Proposition 2.3. Suppose that M 4 is a hypersurface with constant mean curvature H (including H = 0) in a manifold N 5 whose 5 sectional curvature satisfies 17 ≤ K¯ ≤ 1. γ denotes the unit normal vector field of M. If ω ∈ ∧2 (M ), then

⟨E (ω), ω⟩ + Ric(γ , γ )|ω|2 + |A|2 |ω|2 ≥ 0. Proof. Let ω1 = ai1 i2 ei2 ∧ ei1 ∈ ∧2 (M ) and ω2 = bi1 i2 ei2 ∧ ei1 ∈ ∧2 (M ), where ai1 i2 = −ai2 i1 and bi1 i2 = −bi2 i1 . We may compute at a fixed point p. By Lemma 2.1, there is an orthonormal basis {e1 , e2 , e3 , e4 } such that

ω=

n 

αi e2i−1 ∧ e2i ,

i=1

where n ≤ 2. By (2.2), we get that

⟨E (ω), ω⟩ =

n 1

n 1

2 i =1

2 i ,j = 1

(Ric2i−1,2i−1 + Ric2i,2i )αi2 +

(R2i,2j,2j−1,2i−1

− R2i,2j−1,2j,2i−1 − R2i−1,2j−1,2j−1,2i + R2i−1,2j−1,2j,2i )αi αj =

n 1

n 

2 i =1

i,j=1

(Ric2i−1,2i−1 + Ric2i,2i )αi2 −

R2i−1,2i,2j−1,2j αi αj ,

(2.5)

where we make use of the first Bianchi identity in the last equality. By the Gauss equation, we obtain that Rijkl = R¯ ijkl + hik hjl − hil hjk and 4 

Ricii =

(R¯ ikik + hii hkk − h2ik ).

k=1,k̸=i

Combining with (2.5), we have that

⟨E (ω), ω⟩ + Ric(γ , γ )|ω|2 + |A|2 |ω|2   n 4 4   1 2 2 ¯ ¯ = (R2i−1,k,2i−1,k + h2i−1,2i−1 hkk − h2i−1,k ) + (R2i,k,2i,k + h2i,2i hkk − h2i,k ) αi2 2 i =1

−

n 

k=1,k̸=2i−1

k=1,k̸=2i

1

(R¯ 2i−1,2i,2j−1,2j + h2i−1,2j−1 h2i,2j − h2i−1,2j h2j−1,2i )αi αj + Ric(γ , γ ) 2

i,j=1

n 

αi2 + |A|2 |ω|2

i =1

= L1 + L2 ,

(2.6)

where

L1 =

n 1



2 i=1

−

4 

h2i−1,2i−1 hkk −

k=1,k̸=2i−1

n 

h22i−1,k

+



4 

h2i,2i hkk −

k=1,k̸=2i

h22i,k

αi2

(h2i−1,2j−1 h2i,2j − h2i−1,2j h2j−1,2i )αi αj + |A|2 |ω|2

i,j=1

and

L2 =

n 1

2 i=1



4  k=1,k̸=2i−1

R¯ 2i−1,k,2i−1,k +

4  k=1,k̸=2i

 R¯ 2i,k,2i,k

αi2 −

n  i,j=1

R¯ 2i−1,2i,2j−1,2j αi αj +

1 2

Ric(γ , γ )

n  i=1

αi2 .

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P. Zhu / Nonlinear Analysis 75 (2012) 5039–5043

Since ω is a 2-form, we can regard ω as an antisymmetric matrix. By Lemma 2.2, we obtain that

L1 =

n 1

 trAh2i−1,2i−1 −

2 i =1

−

4 

h22i−1,k

+ trAh2i,2i −

k=1

n 

4 

 h22i,k

αi2

k=1

(h2i−1,2j−1 h2i,2j − h2i−1,2j h2j−1,2i )αi αj + |A|2 |ω|2

i,j=1

=

1 2

trA

n 

(h2i−1,2i−1 + h2i,2i )αi2 + 2tr(Aω)2 + 2trA2 ω2 + |A|2 |ω|2

i=1

= −2trA · trAω2 + 2trA2 ω2 + 2tr(Aω)2 + |A|2 |ω|2 ≥ 0.

(2.7)

5 , 1], the curvature tensor of N satisfies [9] Since the sectional curvature K¯ of N 5 is in [ 17

|R¯ ijkl | ≤

8 17

.

Therefore, we get that

L2 ≥

n 1



2 i=1

−

n  i =1

4 

k=1,k̸=2i−1

α + 2 i

5 17

n 10 

17 i=1

4 

+

5

k=1,k̸=2i

α = 2 i

8 17



17 n 

 αi2 −

n 

R¯ 2i−1,2i,2j−1,2j αi αj

i,j=1,i̸=j n 

α − 2 i

i=1

 |αi αj | .

(2.8)

i,j=1,i̸=j

So, if n = 1 or 2, then L2 ≥ 0. Combining with (2.7), we obtain L1 + L2 ≥ 0. The proof is completed.



Proof of Theorem 1.1. Since M 4 ⊂ N 5 is a stable minimal or strongly stable constant mean curvature hypersurface, it implies that I ( h) =



|∇ h|2 − (Ric(γ , γ ) + |A|2 )h2 dv ≥ 0, M

for every h ∈ C0∞ (M ). Choose h = f |ω|, where ω ∈ H 2 (L2 (M )) and f ∈ C0∞ (M ). Then I ( h) = −



f |ω|△(f |ω|) + (Ric(γ , γ ) + |A|2 )f 2 |ω|2 dv M



f |ω|(f 1|ω| + |ω|△f + 2∇ f · ∇|ω|) + (Ric(γ , γ ) + |A|2 )f 2 |ω|2 dv

=− M



f 2 (|ω|1|ω| + Ric(γ , γ )|ω|2 + |A|2 |ω|2 ) +

=− M



f 2 (|ω|1|ω| + Ric(γ , γ )|ω|2 + |A|2 |ω|2 ) −

1 2 1

∇ f 2 · ∇|ω|2 + f △f |ω|2 dv

|ω|2 △f 2 + f △f |ω|2 dv   2 2 2 2 =− f (|ω|1|ω| + Ric(γ , γ )|ω| + |A| |ω| )dv + |ω|2 |∇ f |2 dv M M   =− f 2 F dv + |ω|2 |∇ f |2 dv, =−

M

M

2

M

where the last equality holds because of (2.4) and

F = ⟨E (ω), ω⟩ + Ric(γ , γ )|ω|2 + |A|2 |ω|2 + (|∇ω|2 − |∇|ω| |2 ). Choose f ∈ C0∞ (M ) such that

 0 ≤ f ≤ 1,     r   f ≡ 1 on B , 2 f ≡ 0 on M \ B(r ),     |∇ f | ≤ C , r

P. Zhu / Nonlinear Analysis 75 (2012) 5039–5043

5043

where C is a positive constant. So,



 B( 2r )

F ≤

C2 B(r ) r

|ω|2 = 2

C2 r2

 B(r )

|ω|2 → 0,

(2.9)

as r → +∞. Set

F1 = ⟨E (ω), ω⟩ + Ric(γ , γ )|ω|2 + |A|2 |ω|2 , and

F2 = |∇ω|2 − |∇|ω| |2 . Obviously, F = F1 + F2 . By Proposition 2.3, we obtain that F1 ≥ 0. By the Kato inequality, F2 ≥ 0. Combining with (2.9), we have that F = 0. In particular,

F2 = |∇ω|2 − |∇|ω| |2 = 0. On the other hand, by Lemma 4.2 in [10], we have

|∇ω|2 ≥

3 2

|∇|ω| |2 .

Thus, |ω| is a constant. Since the dimension of N is odd, we obtain that N is of bounded geometry [11,12]. Note that the volume of a complete noncompact minimal (or constant mean curvature) hypersurface M 4 in a manifold of bounded geometry is infinite [13, Corollary 2.1]. Sinceω ∈ H 2 (L2 (M )), we have ω = 0. If ω ∈ H 1 (L2 (M )), by Theorem 1 in [5], then ω = 0. If f ∈ H 0 (L2 (M )), then df = 0. Since M f 2 < +∞ and the volume of M is infinite, f = 0. Note that ω is closed and coclosed for ω ∈ H p (L2 (M )) [14, Proposition 1]. If H p (L2 (M )) = {0}, then H 4−p (L2 (M )) = {0}. In fact, we may use the Hodge operator and reduce L2 harmonic (4 − p)-forms to L2 harmonic p-forms. Since H p (L2 (M )) = {0} for p = 0, 1, 2, we obtain that H p (L2 (M )) = {0} for p = 3, 4. The proof is completed.  References [1] M. do Carmo, C.K. Peng, Stable minimal surfaces in R3 are planes, Bull. Amer. Math. Soc. 1 (1979) 903–906. [2] D. Fischer-Colbrie, R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math. 33 (1980) 199–211. [3] B. Palmer, Stablity of minimal hypersurfaces, Comment. Math. Helv. 66 (1991) 185–188. [4] S. Tanno, L2 harmonic forms and stablity of minimal hypersurfaces, J. Math. Soc. Japan 48 (1996) 761–768. [5] X. Cheng, L2 harmonic forms and stablity of hypersurfaces with constant mean curvature, Bol. Soc. Brasil. Mat. 31 (2000) 225–239. [6] P. Zhu, L2 -harmonic forms and stable hypersurfaces in space forms, Arch. Math. 97 (2011) 271–279. [7] P. Zhu, Harmonic two-forms on manifolds with nonnegative isotropic curvature, Ann. Global Anal. Geom. 40 (2011) 427–434. [8] P. Li, Lecture Notes on Geometric Analysis, in: Lecture Notes Series, vol. 6, Seoul National University, Research Institute of Mathematics, Global Analysis Reseach Center, Seoul, 1993. [9] S. Mendonca, D.T. Zhou, Expression of curvature tensors and some applications, Bol. Soc. Brasil. Mat. 32 (2001) 173–184. [10] X.D. Wang, On the L2 -cohomology of a convex cocompact hyperbolic manifold, Duke Math. J. 115 (2) (2002) 311–327. [11] J. Cheeger, D. Gromoll, On the lower bound for the injectivity radius of 14 -pinched Riemannian manifolds, J. Differential Geom. 15 (3) (1980) 437–442. [12] W. Klingenberg, T. Sakai, Injectivity radius estimate for 14 -pinched manifolds, Arch. Math. 34 (4) (1980) 371–376. [13] X. Cheng, L.F. Cheung, D.T. Zhou, The structure of weakly stable constant mean curature hypersurfaces, Tohoku Math. J. 60 (2008) 101–121. [14] S.T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J. 25 (1976) 659–670.