RIGIDITY OF GROUP ACTIONS

离散群 GAMMA 在可微流形 M 上的平滑作用，根据定义，是 GAMMA 与 Diff M 的同态，即 M 的微分同胚群。群作用理论的主题之一是描绘 GAMMA 在 M 上的平滑作用的整个空间 A (GAMMA，Diff M)。刚性（广义上）是指空间 A(GAMMA，Diff M) 是“小”的主张。不变性几何结构。解决刚性问题的方法之一是寻找在给定的群作用下不变的几何结构。我们找到了一个适合这种方法的新示例：叶流形的全局分析。群体行为的刚性常常相当于叶流形上的一些全局分析问题。我们能够描述某个叶流形上切向拉普拉斯的谱。微分同胚群的无限维齐次空间。我们研究了此类空间的几何和拓扑，特别是考虑到刚性问题的应用。此外，还对无限维空间中的无限维最小子流形（前田）、Yang-Mills梯度流的爆炸现象（前田）、自旋系统的流体动力学极限（铃木）以及某个自由边界问题的正则定理（利根川）进行了研究。

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).