Stability of Hypersurfaces in Minkowsky Normed Spaces

TL;DR

This paper generalizes a classical stability characterization of spheres in Euclidean space to Minkowski spaces, showing that stable, constant mean Minkowski curvature hypersurfaces are homothetic to the boundary of the convex body.

Contribution

It introduces stability concepts and second variation formulas for hypersurfaces in Minkowski spaces, extending classical Euclidean results to this broader setting.

Findings

Stable, constant mean Minkowski curvature hypersurfaces are homothetic to the boundary of the convex body.

Develops second variation formula for surface area measure in Minkowski spaces.

Characterizes stability of hypersurfaces in Minkowski spaces.

Abstract

We extend to Minkowski spaces the classical result of Barbosa and do Carmo [1] that characterizes the euclidean sphere as the unique compact stable CMC hypersurface of $\mathbb R^n$. More precisely, if $K$ is a smooth convex body in $\mathbb R^n$ with positive Gauss curvature, containing the origin in its interior and $M$ is an immersed hypersurface, there are well defined concepts of surface area measure, normal vector field and principal curvatures of $M$ , with respect to $K$. Thus, we introduce the concept of stability with respect to normal variations and compute the formula of second variation with respect to $K$. Finally we show that if $M$ is compact, has constant mean Minkowski curvature and is stable (with respect to $K$) then $M$ is homothetic to $\partial K$.