Uniqueness Problem for Closed Non-smooth Hypersurfaces with Constant Anisotropic Mean Curvature and Self-similar Solutions of Anisotropic Mean Curvature Flow

An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered surface, and it is a generalization of surface area. Equilibrium surfaces with volume constraint are called CAMC (constant anisotropic mean curvature) surfaces and they are not smooth in general. We show that, if the energy density function is two times continuously differentiable and convex, then, like isotropic (constant mean curvature) case, the uniqueness for closed stable CAMC surfaces holds under the assumption of the integrability of the anisotropic principal curvatures. Moreover, we show that, unlike the isotropic case, uniqueness of closed embedded CAMC surfaces with genus zero in the three-dimensional euclidean space does not hold in general. We also give nontrivial self-similar shrinking solutions of anisotropic mean curvature flow. These results are generalized to hypersurfaces in the Euclidean space with general dimension. This article is an announcement of two forthcoming papers by the author.