RIGIDITY OF GROUP ACTIONS

团体行为的刚性

• 批准号: 09640128
• 负责人: KANAI Masahiko
• 金额: $ 1.86万
• 依托单位: Nagoya University (1998)Keio University (1997)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640128/
• 关键词: rigidity of group actions; global analysis on foliated manifolds; 微分同相群の等質空間; スピン系の流体力学極限; 無限次元極小曲面; 化学反応に関する自由境界値問題; 非可控接触代数

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).

根据定义，离散群Gamma在可微流形M上的光滑作用是Gamma到Diff M的同态，M的微分同态群。群作用理论的一个主要主题是刻画Gamma在M上光滑作用的整个空间A(Gamma，diff M)。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。我们发现了一个新的例子，证明了这种方法是可行的。群体行动的刚性常常等同于多叶流形上的某些整体分析问题。我们能够描述某一类分叶流形上的切向拉普拉斯的谱。我们研究了这类空间的几何和拓扑学，特别是应用于头脑中的刚性问题，还研究了无限维空间中的无限维极小子流形(前田)，杨-Mills梯度流的爆破现象(前田)，自旋系统的流体动力学极限(Suzuki)，以及一类自由边界问题的正则性定理(Tonegawa)。

项目成果

M.Kanai: "Rigidity of group actions" Seminaire de Theoris Spectrale et Geometrie, Univ.Grenoble I. 15. (1997)

M.Kanai：“群体行为的刚性”Seminaire de Theoris Spectrale et Geometrie，Univ.Grenoble I. 15。（1997）

H.Kozono, Y.Maeda and H.Naito: "A Yang-Mills-Higgs gradient flow on R^3 blowing up at infinity" Proc.Japan Acad., Ser.A.74. 71-73 (1998)

H.Kozono、Y.Maeda 和 H.Naito：“R^3 上的杨-米尔斯-希格斯梯度流在无穷远处爆炸”Proc.Japan Acad.，Ser.A.74。

Y.Maeda,S.Rosenberg and P.Tondeur: "Minimal orbits of metrics" Journal of Geometry and Physics. 23. 319-349 (1997)

Y.Maeda、S.Rosenberg 和 P.Tondeur：“度量的最小轨道”几何与物理学杂志。

Y.Maeda et al: "Minimal orbits of metrics" J.Geom.Phys.23. 319-349 (1997)

Y.Maeda 等人：“度量的最小轨道”J.Geom.Phys.23。

Y.Tonegawa: "On the regularity of a chemical reaction interface" Comm.Partial Differential Equqtions. 23. 1181-1207 (1998)

Y.Tonekawa：“关于化学反应界面的规律性”Comm.偏微分方程。