Stability of hypersurfaces with constant $r$-th anisotropic mean curvature

TL;DR

This paper investigates the stability of hypersurfaces with constant $r$-th anisotropic mean curvature, demonstrating that the only stable solutions are the Wulff shapes, thus generalizing classical stability results.

Contribution

It introduces a generalized $r$-th anisotropic mean curvature and proves the stability characterization of hypersurfaces with constant curvature as Wulff shapes.

Findings

Wulff shape is the unique stable hypersurface with constant $r$-th anisotropic mean curvature.

Generalization of classical stability results to anisotropic curvature settings.

Establishment of a variational characterization for these hypersurfaces.

Abstract

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, 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. Let $X:M\to \mathbb{R}^{n+1}$ be an $n$-dimensional closed hypersurface with $H^F_{r+1}=$constant, for some $r$ with $0\leq r\leq n-1$, which is a critical point for a variational problem. We show that $X(M)$ is stable if and only if $X(M)$ is the Wulff shape.