Stable hypersurfaces as minima of the integral of an anisotropic mean curvature preserving a linear combination of area and volume

Abstract

We deal with a compact hypersurface $M$ without boundary immersed in Euclidean space $R^{n+1}$ with the quotient of anisotropic mean curvatures $\frac{(r+1)C_{n}^{r+1}H^F_{r+1}}{a(k+1)C_{n}^{k+1}H^F_{k+1}-b}=constant$, for real numbers $a$ and $b$. Such a hypersurface is a critical point for the variational problem preserving a linear combination (with coefficientes $a$ and $b$) of the $(k,F)$-area and the $(n + 1)$-volume enclosed by $M$. We show that $M$ is $(r,k,a,b)$-stable if and only if, up to translations and homotheties, it is the Wulff shape of $F$, under some assumptions on $a$ and $b$ proved to be sharp. For $a=0$ and $b=1$, this gives the known $r$-stability of the $r$-area for volume preserving variations; if also $F\equiv 1$ it yields the stability studied by Alencar-do Carmo-Rosenberg and Barbosa-Colares. For $b=0$ we also prove a characterization of the Wulff shape as a critical point of the $(r,F)$-area for variations preserving the $(k, F)$-area, $0\le k<r<n$, without the $r$-stability hypothesis.