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Geometric Variational Problem and its Applications

Miyuki Koiso
Degree: Doctor of Science (Osaka University)
Research Interests: Differential Geometry
Unit: Space & Flow

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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 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 study by Miyuki Koiso and Bennett Palmer made a new development of this field. Koiso-Palmer obtained many important results about geometric properties, representation formulas, Gauss map and its removable set, existence and uniqueness for stable solutions of free boundary problems, etc.

Rather recently we have 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 with anisotropic surface energy.

My most interest now is constructing a bifurcation theory of solutions of 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.

About education for graduate students, I have been taught minimal surfaces, complex manifolds, geometric variational problems, etc. I have contributed toward developing researchers and specialists of research-education institutes and companies. I would like not only to teach pure mathematics but also to discuss importance of mathematics and mathematical concepts in various fields.

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