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An anisotropic surface energy is the integral of an energy density thatdepends on the normal at each point over the considered surface, and it is ageneralization of surface area. The minimizer of such an energy among allclosed surfaces enclosing the same volume is unique and it is (up to rescaling)so-called the Wulff shape. We prove that, unlike the isotropic case, thereexists an anisotropic energy density function such that there exist closedembedded equilibrium surfaces with genus zero in the three-dimensionalEuclidean space each of which is not (any homothety and translation of) theWulff shape. We also give nontrivial self-similar shrinking solutions ofanisotropic mean curvature flow. These results are generalized to hypersurfacesin the (n+1)-dimensional Euclidean space.

Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature