Stable Minimal Hypersurfaces in the Hyperbolic Space

Seo, Keomkyo

doi:10.4134/jkms.2011.48.2.253

Cited by 16 publications

(22 citation statements)

References 19 publications

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“…( [15], see also [3].) Then the harmonic function u satisfies the inequality (13). Applying the same arguments as in the proof of Theorem 3.4, we conclude that |▽u| = 0.…”

Section: Rigidity Of Minimal Hypersurfacesmentioning

confidence: 62%

“…Theorem ( [13]). Let M be a complete stable minimal hypersurface in H n+1 with finite L 2 -norm of the second fundamental form A (i.e., M |A| 2 dv < ∞).…”

Section: Introductionmentioning

confidence: 99%

“…Let φ be a compactly supported Lipschitz function on M . Put f := |▽u| in the inequality (13). Multiplying both sides of the inequality (13) by φ 2 and integrating over M , we obtain…”

mentioning

confidence: 99%

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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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“…( [15], see also [3].) Then the harmonic function u satisfies the inequality (13). Applying the same arguments as in the proof of Theorem 3.4, we conclude that |▽u| = 0.…”

Section: Rigidity Of Minimal Hypersurfacesmentioning

confidence: 62%

“…Theorem ( [13]). Let M be a complete stable minimal hypersurface in H n+1 with finite L 2 -norm of the second fundamental form A (i.e., M |A| 2 dv < ∞).…”

Section: Introductionmentioning

confidence: 99%

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

Dũng

Seo

2011

Ann Glob Anal Geom

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“…Let C a be the spherical catenoid obtained by rotating σ a ⊂ B 2 + (see (3.3) for the definition of B 2 + ) about the u-axis, where σ a is the catenary given by (3.16) and a > 0 is the hyperbolic distance between σ a and the origin. Mori, Do Carmo and Dajczer, Bérard and Sa Earp, and Seo proved the following result (see [Mor81,dCD83,BSE10,Seo11]): There exist two constants A 1 ≈ 0.46288 and A 2 ≈ 0.5915 such that C a is unstable if 0 < a < A 1 , and C a is globally stable if a > A 2 .…”

Section: Introductionmentioning

confidence: 99%

Stability of catenoids and helicoids in hyperbolic space

Wang

2019

Asian Journal of Mathematics

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In this paper, we study the stability of catenoids and helicoids in the hyperbolic 3-space H 3 . We will prove the following results.(1) For a family of spherical minimal catenoids {Ca}a>0 in the hyperbolicthere exist two constants 0 < ac < a l such that • Ca is an unstable minimal surface with Morse index one if a < ac, • Ca is a globally stable minimal surface if a ac, and • Ca is a least area minimal surface in the sense of Meeks and Yau (see §2.1 for the definition) if a a l . (2) For a family of minimal helicoids {Hā}ā 0 in the hyperbolic space H 3 (see §2.4 for detail definitions), there exists a constantāc = coth(ac) such that • Hā is a globally stable minimal surface if 0 ā āc, and • Hā is an unstable minimal surface with Morse index infinity if a >āc.

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“…The spherical catenoids immersed in H 3 are example of surfaces of revolution in H 3 with transcient (non-recurrent) Brownian movement. Spherical catenoids have been studied in [dCD83], [Mor81] or [Seo11] and, specifically using the Upper Halfspace Model in [BSE10]. and…”

Section: Surfaces In Hmentioning

confidence: 99%

Conformal type of ends of revolution in space forms of constant sectional curvature

Gimeno

Gozalbo

2015

Ann Glob Anal Geom

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ABSTRACT. In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the 3-dimensional Euclidean space or in the 3-dimensional sphere is parabolic. In the case of ends of revolution in the hyperbolic 3-dimensional space, we find sufficient conditions to attain parabolicity for complete ends of revolution using their relative position to the complete flat surfaces of revolution. This paper is concerned with the study of the conformal type (parabolicity or nonparabolicity) of complete ends of revolution immersed in the 3-dimensional Euclidean space R 3 , in the 3-dimensional hyperbolic space H 3 , or in the 3-dimensional sphere S 3 . Let us denote by M 3 (κ) the simply connected space form of constant sectional curvature κ ∈ R. Hence,is a complete end of revolution if there exists a geodesic in M 3 (κ) such that the end is invariant by the group of rotations of M 3 (κ) that leaves this geodesic point-wise fixed. More precisely, an end of revolution will be the rotation along a geodesic ray γ of M 3 (κ) of a generating smooth curve β : [0, ∞) → M 2 (κ) contained in a totally geodesic 2010 Mathematics Subject Classification. Primary 53C20 53C40; Secondary 53C42.

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Stable Minimal Hypersurfaces in the Hyperbolic Space

Cited by 16 publications

References 19 publications

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

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

Stability of catenoids and helicoids in hyperbolic space

Conformal type of ends of revolution in space forms of constant sectional curvature

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