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Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures. (English) Zbl 1166.53035
Summary: Given a positive function $F$ on ${S}^{n}$ which satisfies a convexity condition, for $1\le r\le n$, we define for hypersurfaces in ${ℝ}^{n+1}$ the $r$-th anisotropic mean curvature function ${H}_{r}^{F}$, a generalization of the usual $r$-th mean curvature function. We prove that a compact embedded hypersurface without boundary in ${ℝ}^{n+1}$ with constant ${H}_{r}^{F}$ is the Wulff shape, up to translations and homotheties. In the case $r=1$, our result is the anisotropic version of Alexandrov’s Theorem, which gives an affirmative answer to an open problem of F. Morgan.
##### MSC:
 53C40 Global submanifolds (differential geometry) 53A10 Minimal surfaces, surfaces with prescribed mean curvature 52A20 Convex sets in $n$ dimensions (including convex hypersurfaces)