Open Access
2018 Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space
Jonatan F. da Silva, Henrique F. de Lima, Marco Antonio L. Velásquez
Publ. Mat. 62(1): 95-111 (2018). DOI: 10.5565/PUBLMAT6211805


Given a positive function $F$ defined on the unit Euclidean sphere and satisfying a suitable convexity condition, we consider, for hypersurfaces $M^n$ immersed in the Euclidean space $\mathbb R^{n+1}$, the so-called $k$-th anisotropic mean curvatures $H_k^F$, $0\leq k\leq n$. For fixed $0\leq r\leq s\leq n$, a hypersurface $M^n$ of $\mathbb{R}^{n+1}$ is said to be $(r,s,F)$-linear Weingarten when its $k$-th anisotropic mean curvatures $H_k^F$, $r\leq k\leq s$, are linearly related. In this setting, we establish the concept of stability concerning closed $(r,s,F)$-linear Weingarten hypersurfaces immersed in $\mathbb R^{n+1}$ and, afterwards, we prove that such a hypersurface is stable if, and only if, up to translations and homotheties, it is the Wulff shape of $F$. For $r=s$ and $F\equiv1$, our results amount to the standard stability studied, for instance, by Alencar-do Carmo-Rosenberg [1].


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Jonatan F. da Silva. Henrique F. de Lima. Marco Antonio L. Velásquez. "Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space." Publ. Mat. 62 (1) 95 - 111, 2018.


Received: 5 April 2016; Revised: 6 October 2016; Published: 2018
First available in Project Euclid: 16 December 2017

zbMATH: 06848687
MathSciNet: MR3738184
Digital Object Identifier: 10.5565/PUBLMAT6211805

Primary: 53C42
Secondary: 53B25

Keywords: $(r, s, F)$-linear Weingarten hypersurfaces , $k$-th anisotropic mean curvatures , Euclidean space , stable closed hypersurfaces , Wulff shape

Rights: Copyright © 2018 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.62 • No. 1 • 2018
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