Research of submanifolds in symmetric spaces by usingthe infinite dimensional geometry and the complexification

本研究的主要成果如下：(1)证明了非紧型对称空间中的复等焦子流形在一定条件下与李群作用（Hermann作用）的主轨道全等。(2)证明了紧型对称空间中的非极小等焦子流形沿平均曲率流坍缩为其焦点子流形。(3)研究了Hilbert空间中具有Hilbert Lie群自由作用的不变超曲面的正则化平均曲率流，并证明了该流的一类强凸保存性定理。(4)在一定条件下，对非紧型对称空间上的超极作用进行了分类。

Main results of this research are as follows.(1) We showed that complex equifocal submanifolds in a symmetric space of non-compact type are congruent to principal orbits of Lie group actions called “Hermann action” under certain conditions.(2) We showed that non-minimal equifocal submanifolds in a symmetric space of compact type collapse to their focal submanifolds along the mean curvature flow.(3) We investigated the regularized mean curvature flow for invariant hypersurfaces in a Hilbert space equipped with a Hilbert Lie group free action and proved a certain kind of strongly convex preservability theorem for the flow.(4) We classified hyperpolar actions on symmetric spaces of non-compact type under some conditions.