RIGIDITY OF MINIMAL SUBMANIFOLDS IN HYPERBOLIC SPACE

RIGIDITY OF MINIMAL SUBMANIFOLDS IN HYPERBOLIC SPACE

arXiv:1002.3885v1 [math.DG] 20 Feb 2010

KEOMKYO SEO Abstract. We prove that if an n-dimensional complete minimal submanifold M in hyperbolic space has sufficiently small total scalar curvature then M has only one end. We also prove that for such M there exist no nontrivial L2 harmonic 1-forms on M . Mathematics Subject Classification(2000) : 53C42, 58C40. Key Words and Phrases : minimal submanifold, total scalar curvature, hyperbolic space, L2 harmonic 1-form.

1. Introduction n+1

A complete minimal hypersurface M ⊂ M is said to be stable if the second variation of its volume is always nonnegative for any normal variation with compact support. In [2], Cao, Shen, and Zhu showed that a complete connected stable minimal hypersurface in Euclidean space must have exactly one end. Later Ni[8] proved that if an n-dimensional complete minimal submanifold M in Euclidean R space has sufficiently small total scalar curvature (i.e., M |A|n dv < ∞) then M has only one end. More precisely, he proved Theorem. ([8]) Let M be an n-dimensional complete immersed minimal submanifold in Rn+p , n ≥ 3. If r Z n1 n Cs−1 , |A|n dv < C1 = n − 1 M then M has only one end. (Here Cs is a Sobolev constant in [4].)

Recently the author[9] improved the upper bound C1 of total scalar curvature in the above theorem. In this paper, we shall prove that the analogue of the above theorem still holds in hyperbolic space. Throughout this paper, we shall denote by Hn the n-dimensional hyperbolic space of constant sectional curvature −1. Our main result is the following. Theorem 1.1. Let M be an n-dimensional complete immersed minimal submanifold in Hn+p , n ≥ 5. If the total scalar curvature satisfies q Z n1 1 n(n − 4)Cs−1 , |A|n dv < n−1 M

then M has only one end.

Miyaoka [7] showed that if M is a complete stable minimal hypersurface in Rn+1 , then there are no nontrivial L2 harmonic 1-forms on M . In [12], Yun proved that if Z n1 p M ⊂ Rn+1 is a complete minimal hypersurface with |A|n dv < C2 = Cs−1 , M

then there are no nontrivial L2 harmonic 1-forms on M . Recently the author[9] showed that this result is still true for any complete minimal submanifold M n in 1

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Rn+p , p ≥ 1. We shall prove that if M is an n-dimensional complete minimal submanifold with sufficiently small total scalar curvature in hyperbolic space, then there exist no nontrivial L2 harmonic 1-forms on M . More precisely, we prove Theorem 1.2. Let M be an n-dimensional complete immersed minimal submanifold in Hn+p , n ≥ 5. If s Z n1 n(n − 4) 1 n < |A| dv , (n − 1)2 Cs M then there are no nontrivial L2 harmonic 1-forms on M . 2. Proof of the theorems To prove Theorem 1.1, we begin with the following useful facts. Lemma 2.1 (Sobolev inequality [4]). Let M be an n-dimensional complete immersed minimal submanifold in Hn+p , n ≥ 3. Then for any φ ∈ W01,2 (M ) we have Z Z n−2 2n n n−2 |φ| dv ≤ Cs |∇φ|2 dv, M

M

where Cs depends only on n.

Lemma 2.2. ([5]) Let M be an n-dimensional complete immersed minimal submanifold in Hn+p . Then the Ricci curvature of M satisfies n−1 2 Ric(M ) ≥ −(n − 1) − |A| . n Recall that the first eigenvalue of a Riemannian manifold M is defined as R |∇f |2 R , λ1 (M ) = inf M f f2 M

where the infimum is taken over all compactly supported smooth functions on M . For a complete stable minimal hypersurface M in Hn+1 , Cheung and Leung [3] proved that 1 (2.1) (n − 1)2 ≤ λ1 (M ). 4 Here this inequality is sharp because equality holds when M is totally geodesic ([6]). Let u be a harmonic function on M . From Bochner formula, we have X 1 ∆(|∇u|2 ) = uij 2 + Ric(∇u, ∇u). 2 Then Lemma 2.2 gives X n−1 2 1 ∆(|∇u|2 ) ≥ uij 2 − (n − 1)|∇u|2 − |A| |∇u|2 . 2 n Choose the normal coordinates at p such that u1 (p) = |∇u|(p), ui (p) = 0 for i ≥ 2. Then we have P qX ui uij 2 ∇j |∇u| = ∇j ( ui ) = = u1j . |∇u| P Therefore we obtain |∇|∇u||2 = u1j 2 . On the other hand, we have 1 ∆(|∇u|2 ) = |∇u|∆|∇u| + |∇|∇u||2 . 2


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Therefore it follows X X n−1 2 uij 2 − (n − 1)|∇u|2 − |A| |∇u|2 ≤ |∇u|∆|∇u| + u1j 2 . n Hence we get X X n−1 2 |A| |∇u|2 ≥ uij 2 − u1j 2 |∇u|∆|∇u| + (n − 1)|∇u|2 + n X X ≥ ui1 2 + uii 2 i6=1

≥

X

i6=1

2

ui1 +

i6=1

i6=1

≥ where we used ∆u =

P

1 n−1

1 X ( uii )2 n−1

X i6=1

2

ui1 =

1 |∇|∇u||2 , n−1

uii = 0 in the last inequality. Therefore we get 1 n−1 2 (2.2) |∇u|∆|∇u| + |A| |∇u|2 + (n − 1)|∇u|2 ≥ |∇|∇u||2 . n n−1 Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1. Suppose that M has at least two ends. First we note that if M has more than one end then there exists a nontrivial bounded harmonic function u(x) on M which has finite total energy([11]). Let f = |∇u|. From (2.2) we have n−1 2 2 1 f ∆f + |A| f + (n − 1)f 2 ≥ |∇f |2 . n n−1 Fix a point p ∈ M and for R > 0 choose a cut-off function satisfying 0 ≤ ϕ ≤ 1, 1 ϕ ≡ 1 on Bp (R), ϕ = 0 on M \ Bp (2R), and |∇ϕ| ≤ , where Bp (R) denotes the R ball of radius R centered at p ∈ M . Multiplying both sides by ϕ2 and integrating over M , we have Z Z Z Z 1 n−1 ϕ2 f 2 dv ≥ ϕ2 f ∆f dv + ϕ2 |A|2 f 2 dv + (n − 1) ϕ2 |∇f |2 dv. n n−1 M M M M From the inequality (2.1), we see

(n − 1)2 ≤ λ1 (M ) ≤ 4

R

|∇(ϕf )|2 dv . 2 2 M ϕ f dv

M R

Therefore we get Z Z Z Z 4 1 n−1 2 2 2 2 2 ϕ |A| f dv + |∇(ϕf )| dv ≥ ϕ2 |∇f |2 dv. ϕ f ∆f dv + n n−1 M n−1 M M M

Using integration by parts, we get Z Z Z Z n−1 4 2 2 2 2 2 f ϕh∇f, ∇ϕidv + |∇f | ϕ dv − 2 − ϕ |A| f dv + |∇(ϕf )|2 dv n n−1 M M M M Z 1 ≥ ϕ2 |∇f |2 dv. n−1 M Applying Schwarz inequality, for any positive number a > 0, we obtain Z Z 4 1 4 n−1 + + f 2 |∇ϕ|2 dv ϕ2 |A|2 f 2 dv + n n − 1 a a(n − 1) M M Z n 4 4a (2.3) ϕ2 |∇f |2 dv. ≥ −a− − n−1 n−1 n−1 M

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On the other hand, applying Sobolev inequality(Lemma 2.1), we have Z n−2 Z 2n n −1 2 |∇(f ϕ)| dv ≥ Cs (f ϕ) n−2 dv . M

M

Thus applying Schwarz inequality again, we have for any positive number b > 0, Z Z n−2 2n n −1 2 2 (2.4) ϕ |∇f | dv ≥ Cs (1 + b) (f ϕ) n−2 dv M M Z 1 −(1 + ) f 2 |∇ϕ|2 dv. b M Combining (2.3) and (2.4), we have Z Z n − 4 − 4a n−1 −a ϕ2 |A|2 f 2 dv ≥ ϕ2 |∇f |2 dv n n − 1 M M n−2 Z 2n Cs −1 n − 4 − 4a n ≥ −a (f ϕ) n−2 dv b+1 n−1 M n + 3 + 4a o Z n 1 n − 4 − 4a −a + f 2 |∇ϕ|2 dv. − b n−1 a(n − 1) M

Applying H¨ older inequality, we get Z Z n2 Z n−2 2n n ϕ2 |A|2 f 2 dv ≤ |A|n . (f ϕ) n−2 dv M

M

Finally we obtain n 1 n − 4 − 4a

M

Z n + 3 + 4a o −a + f 2 |∇ϕ|2 dv b n−1 a(n − 1) M n−2 n − 1Z n2 o Z n C −1 n − 4 − 4a 2n n s . (f ϕ) n−2 dv −a − |A|n dv ≥ b+1 n−1 n M M

By the assumption on the total scalar curvature, we choose a and b small enough such that n C −1 n − 4 − 4a n − 1Z n2 o s −a − |A|n dv ≥ε>0 b+1 n−1 n M

for sufficiently small ε > 0. Then letting R → ∞, we have f ≡ 0, i.e., |∇u| ≡ 0. Therefore u is constant. This contradicts the assumption that u is a nontrivial harmonic function. In the proof of Theorem 1.1, if we do not use the fact that λ1 (M ) ≥ assume that n Z n2 n n−1 < Cs −1 − |A|n dv n−1 n − 1 λ1 (M ) M

(n−1)2 4

and

2

, one can see that M n must have exactly one end by using the for λ1 (M ) > (n−1) n same argument as in the above proof, when n ≥ 3. In other words, it follows Theorem 2.3. Let M be an n-dimensional complete immersed minimal submani2 fold in Hn+p , n ≥ 3. Assume that λ1 (M ) > (n−1) and the total scalar curvature n satisfies Z n2 n n n−1 < . |A|n dv Cs −1 − n−1 n − 1 λ1 (M ) M Then M must have only one end.


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Proof of Theorem 1.2. Let ω be an L2 harmonic 1-form on minimal submanifold M in Hn+p . Then ω satisfies Z ∆ω = 0 and |ω|2 dv < ∞. M

It follows from Bochner formula

∆|ω|2 = 2(|∇ω|2 + Ric(ω, ω)). We also have ∆|ω|2 = 2(|ω|∆|ω| + |∇|ω||2 ). Since |∇ω|2 ≥

n |∇|ω||2 by [10], it follows that n−1

|ω|∆|ω| − Ric(ω, ω) = |∇ω|2 − |∇|ω||2 ≥

1 |∇|ω||2 . n−1

By Lemma 2.2, we have |ω|∆|ω| −

n−1 2 2 1 |∇|ω||2 ≥ Ric(ω, ω) ≥ −(n − 1)|ω|2 − |A| |ω| . n−1 n

Therefore we get |ω|∆|ω| +

n−1 2 2 1 |A| |ω| + (n − 1)|ω|2 − |∇|ω||2 ≥ 0. n n−1

Multiplying both sides by ϕ2 as in the proof of Theorem 1.1 and integrating over M , we have from integration by parts that Z 1 n−1 2 2 2 (2.5) 0 ≤ ϕ |A| |ω| + (n − 1)|ω|2 ϕ2 − ϕ2 |∇|ω||2 dv ϕ2 |ω|∆|ω| + n n − 1 M Z Z n ϕ|ω|h∇ϕ, ∇|ω|idv − = −2 ϕ2 |∇|ω||2 dv n − 1 M Z Z M n−1 2 2 + (n − 1) |ω| ϕ dv + |A|2 |ω|2 ϕ2 dv. n M M

On the other hand, we get the following from H¨ older inequality and Sobolev inequality(Lemma 2.1) Z n−2 Z n2 Z 2n n |A|2 |ω|2 ϕ2 dv ≤ (ϕ|ω|) n−2 dv |A|n dv M M M Z n2 Z |∇(ϕ|ω|)|2 dv ≤ Cs |A|n dv M M Z n2 Z = Cs |A|n dv |ω|2 |∇ϕ|2 + |ϕ|2 |∇|ω||2 + 2ϕ|ω|h∇ϕ, ∇|ω|idv . M

M

Then (2.5) becomes (2.6)

Z Z n ϕ2 |∇|ω||2 dv + (n − 1) |ω|2 ϕ2 dv ϕ|ω|h∇ϕ, ∇|ω|idv − n − 1 M M M Z n2 Z n−1 2 2 2 2 n + |ω| |∇ϕ| + ϕ |∇|ω|| + 2ϕ|ω|h∇ϕ, ∇|ω|idv . Cs |A| dv n M M

0 ≤ −2

Z

Applying the inequality (2.1), we have

(n − 1)2 ≤ λ1 (M ) ≤ 4

|∇(ϕ|ω|)|2 dv M R . ϕ2 |ω|2 dv M

R

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Thus Z M

|ω|2 ϕ2 dv ≤

4 (n − 1)2

Z

M

|ω|2 |∇ϕ|2 + ϕ2 |∇|ω||2 + 2ϕ|ω|h∇ϕ, ∇|ω|idv .

Using Schwarz inequality for ε > 0, we have Z Z εZ 2 2 ϕ2 |∇|ω||2 dv + |ω|2 |∇ϕ|2 dv. ϕ|ω|h∇ϕ, ∇|ω|idv ≤ 2 ε M M M

Therefore it follows from the inequality (2.6) " Z n2 n−4 n−1 − Cs |A|n dv n−1 n M # Z n n2 o Z 4 n−1 ε n 1+ + |A| dv Cs − ϕ2 |∇|ω||2 dv 2 (n − 1)2 n M M " Z n2 n−1 4 + Cs |A|n dv ≤ n−1 n M # Z n n2 o Z 2 n − 1 4 n + |ω|2 |∇ϕ|2 dv. + 1+ |A| dv ε (n − 1)2 n M M Z n2 n(n − 4) 1 by assumption, choosing ε > 0 sufficiently < Since |A|n dv (n − 1)2 Cs M small and letting R → ∞, we obtain ∇|ω| ≡ 0, i.e., |ω| is constant. However, since Z |ω|2 dv < ∞ and the volume of M is infinite ([1] and [11].), we get ω ≡ 0. M

In the proof of Theorem 1.1, if we do not use the fact that λ1 (M ) ≥ assume that Z n2 (nλ1 (M ) − (n − 1)2 )n < |A|n dv (n − 1)2 Cs λ1 (M ) M

(n−1)2 4

and

2

for λ1 (M ) > (n−1) , one can see that there exist no nontrivial L2 harmonic 1-forms n n on M for n ≥ 3 by using the same argument as in the above proof. More precisely, we have Theorem 2.4. Let M be an n-dimensional complete immersed minimal submani2 fold in Hn+p , n ≥ 3. Assume that λ1 (M ) > (n−1) and the total scalar curvature n satisfies Z n2 (nλ1 (M ) − (n − 1)2 )n |A|n dv . < (n − 1)2 Cs λ1 (M ) M Then there are no nontrivial L2 harmonic 1-forms on M . References [1] M.T. Anderson, Complete minimal varieties in hyperbolic space, Invent. Math. 69 (1982), no. 3, 477-494. [2] H. Cao, Y. Shen and S. Zhu, The structure of stable minimal hypersurfaces in Rn+1 , Math. Res. Lett. 4 (1997), no. 5, 637-644. [3] L.F. Cheung and P.F. Leung, Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001), no. 3, 525-530. [4] D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715-727. [5] P.F. Leung, An estimate on the Ricci curvature of a submanifold and some applications, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1051-1063.


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[6] H.P. McKean, An upper bound to the spectrum of ∆ on a manifold of negative curvature, J. Diff. Geom. 4 (1970), 359-366. [7] R. Miyaoka, L2 harmonic 1-forms on a complete stable minimal hypersurface, Geometry and Global Analysis. 1993, 289-293. [8] L. Ni, Gap theorems for minimal submanifolds in Rn+1 , Comm. Anal. Geom. 9 (2001), no. 3, 641-656. [9] K. Seo, Minimal submanifolds with small total scalar curvature in Euclidean space, Kodai Math. J. 31 (2008), no. 1, 113-119. [10] X. Wang, On conformally compact Einstein manifolds, Math. Res. Lett. 8 (2001), no. 5-6, 671-685. [11] S.W. Wei, The structure of complete minimal submanifolds in complete manifolds of nonpositive curvature, Houston J. Math. 29 (2003), no. 3, 675-689. [12] G. Yun, Total scalar curvature and L2 harmonic 1-forms on a minimal hypersurface in Euclidean space, Geom. Dedicata 89 (2002), 135-141. Department of Mathematics, Sookmyung Women’s University, Hyochangwongil 52, Yongsan-ku, Seoul, 140-742, Korea E-mail address: [email protected]