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April, 2019 A new stability notion of closed hypersurfaces in the hyperbolic space
Marco Antonio Lázaro VELÁSQUEZ, Henrique Fernandes DE LIMA, Jonatan Floriano DA SILVA, Arlandson Matheus Silva OLIVEIRA
J. Math. Soc. Japan 71(2): 413-428 (April, 2019). DOI: 10.2969/jmsj/78317831


In this article, we establish the notion of strong $(r,k,a,b)$-stability related to closed hypersurfaces immersed in the hyperbolic space $\mathbb{H}^{n+1}$, where $r$ and $k$ are nonnegative integers satisfying the inequality $0 \leq k<r \leq n-2$ and $a$ and $b$ are real numbers (at least one nonzero). In this setting, considering some appropriate restrictions on the constants $a$ and $b$, we show that geodesic spheres are strongly $(r,k,a,b)$-stable. Afterwards, under a suitable restriction on the higher order mean curvatures $H_{r+1}$ and $H_{k+1}$, we prove that if a closed hypersurface into the hyperbolic space $\mathbb{H}^{n+1}$ is strongly $(r,k,a,b)$-stable, then it must be a geodesic sphere, provided that the image of its Gauss mapping is contained in the chronological future (or past) of an equator of the de Sitter space.

Funding Statement

The first author is partially supported by CNPq, Brazil, grant 308757/2015-7. The second author is partially supported by CNPq, Brazil, grant 3003977/2015-9. The fourth author is partially supported by CAPES, Brazil.


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Marco Antonio Lázaro VELÁSQUEZ. Henrique Fernandes DE LIMA. Jonatan Floriano DA SILVA. Arlandson Matheus Silva OLIVEIRA. "A new stability notion of closed hypersurfaces in the hyperbolic space." J. Math. Soc. Japan 71 (2) 413 - 428, April, 2019.


Received: 25 June 2017; Revised: 16 October 2017; Published: April, 2019
First available in Project Euclid: 25 February 2019

zbMATH: 07090049
MathSciNet: MR3943444
Digital Object Identifier: 10.2969/jmsj/78317831

Primary: 53C42
Secondary: 53A10

Keywords: closed hypersurfaces , geodesic spheres , higher order mean curvatures , Hyperbolic space , strong $(r,k,a,b)$-stability

Rights: Copyright © 2019 Mathematical Society of Japan


Vol.71 • No. 2 • April, 2019
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