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

Abstract

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, we define the $r$-th anisotropic mean curvature function $H_r^F$ 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 $H_{r+1}^F=$constant, for some $r$ with $0\le r\le n-1$, 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.

12 pages