Stability of area-preserving variations in space forms

He, Yunze; Li, Haizhong

doi:10.1007/s10455-007-9095-3

Cited by 7 publications

(4 citation statements)

References 13 publications

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“…Volume preserving variational problem is an important theme into the theory of isometric immersion and has been studied by many authors (see, for instance, references [1] to [5,7,12]). It is well known that immersions with constant mean curvature are critical points for the variational problem of minimizing the area functional keeping the balance of volume zero.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

The stability of hypersurfaces revisited

2015

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In this paper, we revisit the problem of characterize (r, s)-stable closed hypersurfaces immersed in a Riemannian space form, which was firstly established in Velásquez et al. (J Math Anal Appl 406:134-146, 2013). With a different approach of that used in the proof of the main theorem of Velásquez et al. (J Math Anal Appl 406:134-146, 2013), we complete its program showing that a closed hypersurface contained in the Euclidian space R n+1 and having higher order mean curvatures linearly related is (r, s)-stable if, and only if, it is a geodesic sphere of R n+1 .

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

The stability of hypersurfaces revisited

2015

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“…In this setting, we establish a similar result to Proposition 5.1 of [9] and Proposition 5.3 of [16]. Next, for nonnegative real numbers a 0 and a 1 (with at least one nonzero),…”

Section: Strongly Stable Linear Weingarten Hypersurfaces In H N+1mentioning

confidence: 66%

“…They showed that such hypersurfaces are r-stable if, and only if, they are geodesic spheres; thus generalizing the previous results on constant mean curvature closed hypersurfaces. Later on, He and Li [16] treated the case of compact hypersurfaces without boundary immersed in space forms with the quotient H r+1 /H constant, for some 1 ≤ r ≤ n − 1. They proved that such hypersurfaces are r-stable if, and only if, they are totally umbilical.…”

Section: Introductionmentioning

confidence: 99%

Strongly Stable Linear Weingarten Hypersurfaces Immersed in the Hyperbolic Space

2015

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In this article, we establish the notion of strong stability related to closed linear Weingarten hypersurfaces immersed in the hyperbolic space. In this setting, initially we show that geodesic spheres are strongly stable. Afterwards, under a suitable restriction on the mean and scalar curvatures, we prove that if a closed linear Weingarten hypersurface into the hyperbolic space is strongly stable, then it must be a geodesic sphere, provided that the image of its Gauss mapping is contained in a chronological future (or past) of the de Sitter space.Mathematics Subject Classification. Primary 53C42; Secondary 53B30, 53C50, 53Z05, 83C99.

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“…The stability of hypersurfaces for volume preserving variational problem has a long history since the first result for the stability of constant mean curvature in the Euclidean space by Barbosa and do Carmo [3]. Many authors have been interested in stability problems in various contexts, like for other space forms and/or higher order mean curvatures (see [1,2,4,5,7,8,9,10,12] for instance). In 2013, Velásquez, de Sousa and de Lima [21] defined the notion of (r, s)-stability which generalizes the classical notion of stability for the mean curvature or r-stability for higher order mean curvatures.…”

Section: Introductionmentioning

confidence: 99%

On the stability of the Wulff shape

Roth¹

2015

Preprint

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Given a positive function F on S n satisfying an appropriate convexity assumption, we consider hypersurfaces for which a linear combination of some higher order anisotropic curvatures is constant. We define the variational problem for which these hypersurfaces are critical points and we prove that, up to translations and homotheties, the Wulff shape is the only stable closed hypersurface of the Euclidean space for this problem.

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Stability of area-preserving variations in space forms

Abstract: In this article, we deal with compact hypersurfaces without boundary immersed in space forms with S r+1 S 1 = constant. They are critical points for an area-preserving variational problem. We show that they are r -stable if and only if they are totally umbilical hypersurfaces.

Cited by 7 publications

References 13 publications

The stability of hypersurfaces revisited

The stability of hypersurfaces revisited

Strongly Stable Linear Weingarten Hypersurfaces Immersed in the Hyperbolic Space

On the stability of the Wulff shape

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