Stability of Hypersurfaces in Minkowsky Normed 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 $\partial K$.