Eigenvalue estimates for minimal surfaces in hyperbolic space

Candel, Alberto

doi:10.1090/s0002-9947-07-04104-9

Cited by 18 publications

(12 citation statements)

References 17 publications

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“…Throughout this paper, we shall denote by H n the n-dimensional hyperbolic space of constant sectional curvature −1. Recently Candel [2] gave an upper bound for the first eigenvalue of the universal cover of a complete stable minimal surface in H 3 . Indeed, he proved:…”

Section: Introductionmentioning

confidence: 99%

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Stable Minimal Hypersurfaces in the Hyperbolic Space

Seo¹

2011

Journal of the Korean Mathematical Society

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Abstract. In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has finite L 2 -norm of the second fundamental form on M . We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.

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Section: Introductionmentioning

confidence: 99%

“…Theorem ( [2]). Let Σ be a complete simply connected stable minimal surface in the 3-dimensional hyperbolic space.…”

Section: Introductionmentioning

confidence: 99%

Stable Minimal Hypersurfaces in the Hyperbolic Space

Seo¹

2011

Journal of the Korean Mathematical Society

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“…Candel [2] gave an upper bound for the first eigenvalue of the universal cover of a complete stable minimal surface in the 3-dimensional hyperbolic space H 3 . More precisely, it was proved Theorem ( [2]).…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, it was proved Theorem ( [2]). Let M be a complete simply connected stable minimal surface in H 3 .…”

Section: Introductionmentioning

confidence: 99%

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

Dũng

Seo

2011

Ann Glob Anal Geom

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We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L 2 harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.Mathematics Subject Classification(2000) : 53C42, 58C40.

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“…A well studied problem in the geometry of the Laplacian is the relation between the first eigenvalue/fundamental tone of open sets of a Riemannian manifold and its geometric invariants; see [2], [3], [8] and the references therein. Another interesting problem is to give bounds for the the first eigenvalue/fundamental tone of open sets of minimal submanifolds of Riemannian manifolds; see [4], [5], [7], [9], [10]. There has recently been increasing interest in the study of minimal surfaces (constant mean curvature) in product spaces N × R, after the discovery of many beautiful examples in those spaces; see [16], [17].…”

Section: Introductionmentioning

confidence: 99%