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 $1\leq r\leq n$, 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2007
- DOI:
- 10.48550/arXiv.0712.0694
- arXiv:
- arXiv:0712.0694
- Bibcode:
- 2007arXiv0712.0694H
- Keywords:
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- Mathematics - Differential Geometry;
- 53C40 (Primary);
- 53A10;
- 52A20 (Secondary)
- E-Print:
- 15 pages