M athematical I nequalities & A pplications Volume 15, Number 1 (2012), 69–75

EIGENVALUE ESTIMATES FOR STABLE MINIMAL HYPERSURFACES

K EOMKYO S EO

(Communicated by H. Martini) Abstract. In this article we provide estimates on the first eigenvalue of stable minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded from below and above by negative constants. We also obtain a lower bound of the total scalar curvature of a stable minimal hypersurface if the scalar curvature of the ambient space is positive.

1. Introduction As it is well known, the first Dirichlet eigenvalue of a Riemannian manifold Σ with boundary is characterized as |∇ f |2 λ1 = inf ΣR 2 , f Σ f R

where the infimum is taken over all piecewise smooth functions in Σ vanishing on the boundary ∂ Σ. Recently Candel [1] gave an upper bound for the first eigenvalue of a stable minimal surface in H3 . We say that a minimal hypersurface Σ in an (n + 1)dimensional Riemannian manifold M is stable if the second variation of its volume is always nonnegative for every compactly supported deformation of Σ in M n+1 . More precisely, an n -dimensional minimal hypersurface Σ in a Riemannian manifold M is called stable if for any compactly supported Lipschitz function f on M Z

Σ

|∇ f |2 − (Ric(en+1 ) + |A|2) f 2 > 0

(1.1)

holds, where Ric(en+1 ) is the Ricci curvature of M in the direction of en+1 , en+1 is the unit normal vector of Σ in M , and |A|2 is the squared length of the second fundamental form of Σ. Recall that the Yamabe invariant of the conformal class [g] of an n -dimensional Riemannian manifold Σ is defined by 2n

Y (g) = inf{E(g) : g = u(x) n−2 g, u(x) > 0, u ∈ H 1 (M)}, Mathematics subject classification (2010): 53C21, 53A10. Keywords and phrases: Stability, minimal hypersurface, first eigenvalue, Yamabe invariant. This research was supported by the Sookmyung Women’s University Research Grants 2009. c D l , Zagreb

Paper MIA-15-07

69

70

K EOMKYO S EO

where E(g) is defined by E(g) =

R

4(n−1) 2 2 Σ ( n−2 |∇u| + Rg u )dvg

R

(

Σu

2n n−2

dvg )

n−2 n

.

Here Rg and dvg denote the scalar curvature and the volume form of the metric g on Σ, respectively. Then the Yamabe invariant of Σ is defined by

σ (Σ) = sup{Y (g) : g is a smooth metric on Σ }. Ho [4] generalized Candel’s result to higher-dimensional cases. He gave estimates on the first eigenvalue of a stable minimal hypersurface Σ in hyperbolic space Hn+1 for n > 3 under the assumption that the Yamabe invariant σ (Σ) satisfies that σ (Σ) < 0 . Actually he proved T HEOREM 1.1. Let Σ ⊂ Hn+1 be a compact stable minimal hypersurface with boundary. If σ (Σ) < 0 , then the first eigenvalue of Σ satisfies n2 (n − 2) 1 (n − 1)2 6 λ1 (Σ) < . 4 7n − 6 In this paper we extend Ho’s result to stable minimal hypersurfaces in a Riemannian manifold of variable curvature. We provide the upper bound of the first eigenvalue when the ambient space has negative scalar curvature. More precisely, we prove T HEOREM 1.2. Let M be an (n + 1)-dimensional Riemannian manifold with scalar curvature S satisfying that −S0 6 S < 0 for some positive constant S0 , n > 3 . Let Σ ⊂ M be a compact stable minimal hypersurface with boundary. If σ (Σ) < 0 , then the first eigenvalue of Σ satisfies

λ1 (Σ) <

S0 (n − 2) . 2n

If the ambient space M has sectional curvature bounded from below and above by negative constants, we can extend Theorem 1.1 as follows. T HEOREM 1.3. Let M be an (n + 1)-dimensional Riemannian manifold with sectional curvature KM satisfying −b 6 KM 6 −a < 0 for some positive constants a and b , n > 3 . Let Σ ⊂ M be a compact stable minimal hypersurface with boundary. Assume 1 that σ (Σ) < Cσ for some constant Cσ ∈ [0, 3n−2 n−2 Cs ), where Cs is a Sobolev constant in [5]. Then the first eigenvalue of Σ satisfies (n − 1)2 n(n + 1)b − na a 6 λ1 (Σ) < 3n−2 . 4 n−2 − Cσ Cs When the scalar curvature of the ambient space is positive, we estimate the total scalar curvature of a stable minimal hypersurface Σ and the Yamabe invariant σ (Σ) of Σ.

E IGENVALUE ESTIMATES FOR STABLE MINIMAL HYPERSURFACES

71

T HEOREM 1.4. Let M be an (n + 1)-dimensional compact Riemannian manifold with scalar curvature S > n(n + 1)k > 0 for some positive constant k . Let Σ ⊂ M be a compact Z stable minimal hypersurface without boundary. Then we have the following. (i)

Σ

R > n(n + 1)kVol(Σ) , where R is the scalar curvature of Σ.

(ii) σ (Σ) > 0 , when n > 3 . We note that the Yamabe invariant σ (Σ) is positive if and only if Σ admits a metric of positive scalar curvature. (See [6], [7] and [9].) 2. Proof of the theorems Let M be an (n + 1)-dimensional Riemannian manifold and let Σ be a stable minimal hypersurface in M . Choose an orthonormal frame {e1 , · · · , en , en+1 } adapted to M , so that e1 , · · · , en are tangential and en+1 is the unit normal vector. For 1 6 i, j, k, l 6 n + 1 , let Ri jkl denote the curvature tensor of M . For 1 6 i, j, k, l 6 n , let Ki jkl denote the curvature tensor of Σ with respect to the induced metric from M . For 1 6 i, j 6 n , let hi j = −h∇ei en+1 , e j i be the second fundamental form of Σ, where ∇ is the Riemannian connection on M . Then the Gauss curvature equation says Ki ji j = Ri ji j + hii h j j − h2i j

(2.1)

for 1 6 i, j 6 n . Summing (2.1), we get n

n



Ki ji j =



Ri ji j +

i, j=1

i, j=1



n

∑ hii

i=1

2

n



h2i j .



i, j=1

Since ∑ni=1 hii = 0 by the minimality of Σ, the scalar curvature S of M is



i, j=1

n

n

n+1

S=

Ri ji j = 2 ∑ Rn+1,i,n+1,i + i=1



Ri ji j



h2i j

i, j=1 n

= 2Ric(en+1 ) + R +

(2.2)

i, j=1

= 2Ric(en+1 ) + R + |A|2, where R is the scalar curvature of Σ. Putting this into the stability inequality (1.1), we therefore get 1 2

Z

Σ

Sf2 −

1 2

Z

Σ

Rf2 +

1 2

Z

Σ

|A|2 f 2 6

Z

Σ

|∇ f |2

(2.3)

for any compactly supported smooth function f defined on Σ. Proof of Theorem 1.2. By the inequality (2.3), for any compactly supported function f , we have Z

Σ

2

Sf > 2

Z

2

Σ

|∇ f | +

Z

Σ

R f 2.

(2.4)

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K EOMKYO S EO

Since σ (Σ) < 0 , Y (g) < 0 for the induced metric g on Σ ⊂ M . Thus there exists a 2n smooth function f on Σ such that E( f n−2 g) < 0 , which implies Z

Σ

Rf2 +

4(n − 1) n−2

Z

Σ

|∇ f |2 < 0.

(2.5)

Combining (2.4) and (2.5), we have 4(n − 1) R f < 2 |∇ f | − n−2 Σ Σ

Z

Z

2

2

Z

Σ

|∇ f |2 .

Thus the curvature assumption on M gives 2n n−2

Z

Σ

|∇ f |2 < −

Z

Σ

S f 2 6 S0

Z

f 2,

Σ

which implies |∇ f |2 S0 (n − 2) .  λ1 (Σ) 6 ΣR 2 < 2n f Σ R

Before proving Theorem 1.3, we need the following Sobolev inequality. L EMMA 2.1. ([5]) Let Σ be an n -dimensional complete immersed minimal submanifold in a Riemannian manifold M with nonpositive sectional curvature, n > 3 . Then for any φ ∈ W01,2 (M) we have Z

2n

Σ

|φ | n−2 dv

 n−2 n

6 Cs

Z

Σ

|∇φ |2 dv,

where Cs depends only on n . In [10], the author recently proved a lower bound part in Theorem 1.3. However, for completeness we shall give this part again here. Proof of Theorem 1.3. First we estimate a lower bound of λ1 (Σ). The Laplacian of the distance function r on Σ ⊂ M satisfies √ √ √ ∆r > a(n − |∇r|2 ) coth ar > (n − 1) a, see [3]. Integrating both sides over a domain Ω ⊂ Σ, we get Z Z √ (n − 1) aArea(Ω) 6 ∆rdv =

∂r ds 6 Length(∂ Ω). ∂Ω ∂ ν



(2.6)

Recall that for a Riemannian manifold Σ, the Cheeger constant h(Σ) is defined by h(Σ) := inf Ω

Length(∂ Ω) , Area(Ω)

E IGENVALUE ESTIMATES FOR STABLE MINIMAL HYPERSURFACES

73

where Ω ranges over all open submanifolds of Σ with compact closure in Σ. Then, applying Cheeger’s inequality [2] and inequality (2.6), we obtain (n − 1)2 1 a, λ1 (Σ) > h(Σ)2 = 4 4

(2.7)

which gives the proof for the lower bound part. Now we prove the upper bound part of the first eigenvalue λ1 (Σ). Since Σ is stable, we have Z

Σ

(Ric(en+1 ) + |A|2) f 2 6

Z

Σ

|∇ f |2

for any compactly supported smooth function f on Σ. Using the equation (2.2), we get Z

Σ

Sf2−

Z

Σ

Rf2 −

Z

Σ

Ric(en+1 ) f 2 6

Z

Σ

|∇ f |2 .

From the curvature assumption on M , we have −n(n + 1)b 6 S 6 −n(n + 1)a < 0 and

−nb 6 Ric(en+1 ) = Rn+1,1,n+1,1 + · · · + Rn+1,n,n+1,n 6 −na < 0. Hence −n(n + 1)b

Z

f 2 + na

Σ

Z

f2 6

Σ

Z

Σ

Rf2 +

Z

Σ

|∇ f |2 .

(2.8)

Since σ (Σ) < Cσ , Y (g) < Cσ for the induced metric g on Σ. Then there exists a smooth function f satisfying R

ΣRf

R 2 + 4(n−1) 2 Σ |∇ f | n−2

R

Σ

f

2n n−2

 n−2 n

< Cσ .

Applying Sobolev’s inequality (Lemma 2.1), we have Z

Σ

Rf2+

4(n − 1) n−2

Z

Σ

|∇ f |2 < Cσ Cs

Z

Σ

|∇ f |2 .

(2.9)

Combining the inequalities (2.8) and (2.9), we obtain  3n − 2 n−2

− Cσ Cs

Z

Σ

|∇ f |2 < (n(n + 1)b − na)

Note that the assumption on Cσ implies that that

3n−2 n−2

f 2.

Σ

− Cσ Cs > 0 . Therefore it follows

n(n + 1)b − na |∇ f |2 λ1 (Σ) 6 ΣR 2 < 3n−2 , f Σ n−2 − Cσ Cs R

Z

74

K EOMKYO S EO

which completes the proof of the upper bound part.



In particular, when a = b = 1 , the ambient space M is isometric to the hyperbolic space Hn+1 . As a consequence of Theorem 1.3, we improve the upper bound of the Yamabe invariant in Theorem 1.1 as follows. C OROLLARY 2.2. Let Σ ⊂ Hn+1 be a compact stable minimal hypersurface. As1 sume that σ (Σ) < Cσ for some constant Cσ ∈ [0, 3n−2 n−2 Cs ), where Cs is a Sobolev constant as in Theorem 1.3. Then the first eigenvalue of Σ satisfies (n − 1)2 6 λ1 (Σ) < 4

n2 . 3n−2 n−2 − Cσ Cs

R EMARK 2.3. If Cσ = 0 , then this result is exactly the same as Theorem 1.1. Note that our scalar curvature is exactly twice the scalar curvature in Ho’s paper [4]. Proof of Theorem 1.4. The inequality (2.3) says that 1 1 S f 2 6 |∇ f |2 + Rf2 2 Σ 2 Σ Σ for any compactly supported smooth function f defined on Σ. Since S > n(n + 1)k > 0 by assumption, we have Z

Z

Z

n(n + 1)k 1 f 2 6 |∇ f |2 + 2 2 Σ Σ Choosing a test function f ≡ 1 on Σ gives Z

Z

n(n + 1)k

Z

S6

Σ

Z

Z

R f 2.

(2.10)

Σ

R,

Σ

which completes the proof of (i). For (ii), suppose that σ (Σ) 6 0 . Then there exists a smooth function f > 0 on Σ satisfying that 4(n − 1) |∇ f |2 6 0. (2.11) n−2 Σ Σ Then by (i) it is easy to see that f cannot be constant. Using the inequalities (2.10) and (2.11), we get Z

0 < n(n + 1)k

Z

Σ

f2 6 2

Rf2 +

Z

Σ

|∇ f |2 −

Z

4(n − 1) n−2

Z

Σ

|∇ f |2 = −

which is a contradiction. Therefore we see that σ (Σ) > 0 .

2n n−2

Z

Σ

|∇ f |2 < 0,



R EMARK 2.4. In particular, when n = 2 , it follows from the above theorem that > 0 , where KΣ is the Gaussian curvature of Σ. By the Gauss-Bonnet theorem, Σ cannot have positive genus, which is a result of Schoen-Yau [8]. R

Σ KΣ

Acknowledgement. The author would like to thank the referee for the helpful comments and suggestions.

E IGENVALUE ESTIMATES FOR STABLE MINIMAL HYPERSURFACES

75

REFERENCES [1] A. C ANDEL, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc., 359 (2007), 3567–3575. [2] I. C HAVEL, Isoperimetric inequalities. Differential geometric and analytic perspectives, Cambridge Tracts in Mathematics 145, Cambridge University Press, Cambridge, 2001. [3] J. C HOE, The isoperimetric inequality for minimal surfaces in a Riemannian manifold, J. Reine Angew. Math., 506 (1999), 205–214. [4] P.T. H O, Eigenvalue estimate for minimal hypersurfaces in hyperbolic space, Differential Geom. Appl., 27 (2009), 104–108. [5] D. H OFFMAN AND J. S PRUCK, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math., 27 (1974), 715–727. [6] J. K AZDAN AND F. WARNER, Prescribing curvatures, Proc. Sympos. Pure Math., Vol. 27 (1975), 309–319. [7] R. S CHOEN, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Lecture Notes in Math. 1365, Springer, 1989, 120–154. [8] R. S CHOEN AND S.-T. YAU, Existence of incompressible minimal surfaces and the topology of threedimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2), 110 (1979), 127–142. [9] R. S CHOEN AND S.-T. YAU, On the structure of manifolds with positive scalar curvature, Manuscripta Math., 28 (1979), 159–183. [10] K. S EO, Stable minimal hypersurfaces in the hyperbolic space, to appear in J. Korean Math. Soc.

(Received November 18, 2009)

Mathematical Inequalities & Applications

www.ele-math.com [email protected]

Keomkyo Seo Department of Mathematics Sookmyung Women’s University Hyochangwongil 52, Yongsan-ku Seoul, 140-742, Korea e-mail: [email protected]

EIGENVALUE ESTIMATES FOR STABLE MINIMAL ...

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