Room 440, Astronomy-Mathematics Building, NTU

(台灣大學天文數學館 440室)

Anisotropic surface energy and surfaces with edges

Miyuki Koiso (Kyushu University)

Abstract:

We study variational problems for surfaces with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density which depends on the surface normal over the considered surface. It was introduced by Gibbs to model the equilibrium shape of a small crystal. If the energy density is constant one, the anisotropic surface energy is the usual area of the surface. Surfaces with constant anisotropic mean curvature (CAMC surfaces) are equilibrium surfaces of a given anisotropic surface energy functional for volume-preserving variations. They are generalizations of minimal (and maximal) surfaces and surfaces with constant mean curvature in the Euclidean space and in the Lorentz-Minkowski space. The minimizer of an anisotropic surface energy among all closed surfaces enclosing the same volume is called the Wulff shape. These concepts are naturally generalized to higher dimensions, and they have many applications inside and outside mathematics. In general, the Wulff shape and the equilibrium surfaces are not smooth. In this talk, we give fundamental geometric and analytic properties of CAMC hypersurfaces and recent progress in the research on the uniqueness of closed CAMC hypersurfaces with and without edges.