Curvaturas médias anisotrópicas : estabilidade e resultados para hipersuperfícies não-convexas

Abstract

This work consists of two parts. In the first part we deal with a compact hypersurface without boundary immersed in to the Euclidean space with the quotient of anisotropic mean curvatures constant. Such a hypersurface is a critical point for the variational problem preserving a linear combination of the (k,F)-area and (n + 1)-volume enclosed by M. We show that it is (r, k,a,b)-stable if, and only if, up to translations and homotheties, it is the Wulff shape, under some assumptions on a,b € R. In the second part we obtain further characterizations for the Wulff shape involving the anisotropic mean curvatures of higher order of a hypersurface M in Rn+1 and the set W = Rn+1-Up€M Tp. Results are obtained for non-convex compact hypersurfaces satisfying W ╪ Ø.