Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

Details Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

2007-12-05

arxiv.org/abs/0712.0694v2

Mathematics

Differential Geometry

15 pages

Scientific paper

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, 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. We prove that a compact embedded hypersurface without boundary in $\R^{n+1}$ with $H^F_r={constant}$ 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.

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