Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

Abstract

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, for $1\le r\le n$, 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. We prove that a compact embedded hypersurface without boundary in ${\mathbb{R}}^{n+1}$ with $H_r^F=constant$ is the Wulff shape, up to translations and homotheties. In case $r=1$, our result is the anisotropic version of Alexandrov Theorem, which gives an affirmative answer to an open problem of F. Morgan.

15 pages