Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature

Abstract

We study a variational problem for piecewise-smooth hypersurfaces in the $(n + 1)$-dimensional Euclidean space. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface, which is a generalization of the area of surfaces. The minimizer of such an energy among all closed hypersurfaces enclosing the same $(n + 1)$-dimensional volume is unique and it is (up to rescaling) so-called the Wulff shape. The Wulff shape and equilibrium hypersurfaces of this energy for volume-preserving variations are not smooth in general. In this article we give recent results on the uniqueness and non-uniqueness for closed equilibria. We also give nontrivial self-similar shrinking solutions of anisotropic mean curvature flow. This article is an announcement of forthcoming papers [10], [13].