Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

TL;DR

This paper proves that compact embedded hypersurfaces with constant higher order anisotropic mean curvature are Wulff shapes, extending Alexandrov's theorem to anisotropic settings and solving an open problem by F. Morgan.

Contribution

It establishes that hypersurfaces with constant higher order anisotropic mean curvature are necessarily Wulff shapes, generalizing classical results to anisotropic curvature functions.

Findings

Hypersurfaces with constant $H^F_r$ are Wulff shapes.

Extends Alexandrov's theorem to anisotropic curvature.

Provides an affirmative answer to F. Morgan's open problem.

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.