Stability of hypersurfaces with constant $r$-th anisotropic mean curvature

Details Stability of hypersurfaces with constant $r$-th anisotropic mean curvature Stability of hypersurfaces with constant $r$-th anisotropic mean curvature

2008-01-23

arxiv.org/abs/0801.3561v1

Mathematics

Differential Geometry

12 pages

Scientific paper

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 $H^F_{r+1}=$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.

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