GCOE ホームページにおける自己紹介文 (英語版)　2011.3.25小磯　深幸 (Miyuki KOISO)

Title: Global analysis of variational problems for surfaces

Geometric variational problem is one of the most fundamental subjects of differential geometry. I am working on mainly variational problems for hypersurfaces in Riemannian manifolds with constant curvature. In general, a solution of a variational problem is said to be stable if the second variation of the energy is nonnegative. Especially, a solution which attains a minimum of the energy is stable. Therefore, it is important to study stability for solutions from both mathematical and physical point of view. My main interests are existence, uniqueness, stability, and global properties of solutions. My recent subjects of study are surfaces with constant mean curvature (CMC surfaces) and surfaces with constant anisotropic mean curvature (CAMC surfaces). The former are critical points of area (isotropic surface energy) for volume-preserving variations, while the latter are critical points of anisotropic surface energy for volume-preserving variations. So, CMC surfaces serve as mathematical models of thin liquid bubbles, and CAMC surfaces serve as mathematical models of, for example, certain kinds of small liquid crystals. Usually, CMC surfaces are regarded as a special case of CAMC surfaces. CMC surface is a classical subject, and it is studied still now very actively. Also, these days CAMC surfaces are studied in many research areas of not only mathematics but also other fields, for example, physics, technology, as both basic research and applied science. Although there had not been a lot of geometric research of CAMC surfaces until rather recently, a series of joint study by Professor Bennett Palmer (Idaho State University) and myself made a new development of this field. Koiso-Palmer obtained many important results about geometric properties, representation formulas, Gauss map (the unit normal) and its removable set of CAMC surfaces, existence and uniqueness for stable solutions of free boundary problems for anisotropic surface energies, etc. Rather recently we proved that any CAMC surface which is a topological sphere is a rescaling of the Wulff shape. We expect that this result can contribute toward determining the shape of materials which have anisotropic surface energies. At present, my greatest interest is constructing a bifurcation theory of solutions of geometric variational problems with constraint. Especially, it is interesting to study bifurcations from stable solutions with symmetry to stable solutions with less symmetry, which may be important from both mathematical and physical point of view. I would like to emphasize that the

problem of stability for variational problems with constraint is much more complicated than that for variational problems without constraint, and variational problems with constraint appear naturally in various situations. Actually, CMC and CAMC surfaces are such examples. I hope that this research will not only contribute development of mathematics, but also give mathematical assurance of various physical phenomena.