RIGIDITY OF GROUP ACTIONS

点击翻译按钮获取中文摘要

A smooth action of a discrete group GAMMA on a differentiable manifold M is, by definition, a homomorphism of GAMMA into Diff M, the diffeomorphism group of M.One of the main theme in the theory of group actions is to depict the whole space A (GAMMA, Diff M) of smooth actions of GAMMA on M.By rigidity (in a wide sense) is meant a claim which says that the space A(GAMMA, Diff M) is "small".Invariant Geometric Structures. One of the approaches to the rigidity problem is frying to find a geometric structure that is invariant under a given group action. We found a new example for which this approach works.Global Analysis on Foliated Manifolds. Rigidity for group actions often amounts to some global-analytic problem on a foliated manifold. We were able to describe the spectrum of the tangential laplacian on a certain foliated manifold.Infinite-Dimensional Homogeneous Spaces of Diffeomorphism Groups. We studied geometry and topology of such spaces especially bearing an application to rigidity problem in mind.Also there have been done researches on infinite-dimensional minimal submanifolds in infinite-dimensional spaces (by Maeda) , on the blow-up phenomenon of the Yang-Mills gradient flow (by Maeda) , on hydrodynamic limit of a spin system (by Suzuki) , and on a regularity theorem for a certain free boundary problem (by Tonegawa).