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.
Citation
Yijun He. Haizhong Li. "Stability of hypersurfaces with constant $(r+1)$-th anisotropic mean curvature." Illinois J. Math. 52 (4) 1301 - 1314, Winter 2008. https://doi.org/10.1215/ijm/1258554364
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