Illinois Journal of Mathematics

Stability of hypersurfaces with constant $(r+1)$-th anisotropic mean curvature

Yijun He and Haizhong Li

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Abstract

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. Let $X : M\to\mathbb{R}^{n+1}$ be an $n$-dimensional closed hypersurface with $H^F_{r+1}=\mathrm{constant}$, for some $r$ with $0\leq r\leq n-1$, which is a critical point for a variational problem. We show that $X(M)$ is stable if and only if $X(M)$ is the Wulff shape.

Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1301-1314.

Dates
First available in Project Euclid: 18 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258554364

Digital Object Identifier
doi:10.1215/ijm/1258554364

Mathematical Reviews number (MathSciNet)
MR2595769

Zentralblatt MATH identifier
1181.53052

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 49Q10: Optimization of shapes other than minimal surfaces [See also 90C90]

Citation

He, Yijun; Li, Haizhong. Stability of hypersurfaces with constant $(r+1)$-th anisotropic mean curvature. Illinois J. Math. 52 (2008), no. 4, 1301--1314. doi:10.1215/ijm/1258554364. https://projecteuclid.org/euclid.ijm/1258554364


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