Hermitian对称空间中子流形的平均曲率流

This project aims to apply mean curvature flow to study the properties of Hermitian symmetric spaces and the submanifolds in them . Suppose that the second fundamental form of the submanifold satisfies a pinching condition, we expect that it is invariant under the mean curvature flow. We give an effiective estimation of the reaction terms in the evolution equations, then we analyze the existence and convergence of the solution. Thus we obtain the topology of the submanifolds. This is a generalization of the work of G. Pipoli and C. Sinestrari about the closed submanifods in complex projective spaces. Moreover we will apply mean curvature flow to study the deformations of symplectomorphisms of all kinds of irreducible Hermitian symmetric spaces of compact type and their submanifolds. We will use the method of bounded complex symmetric domains to deal with the types of U(n + m)/U(n) × U(m), n, m ≥ 1, SO(n + 2)/SO(n) × SO(2), n ≥ 3. This is a supplement of the work of Guangcun Lu and Bang Xiao on this area.

本课题旨在应用平均曲率流来研究Hermitian对称空间及其子流形的相关性质。假设此子流形的第二基本形式满足适当的pinching条件，比如说它的模长可以用平均曲率的模长来界定。我们期望这个pinching条件在曲率流发展的过程中得以保持。通过有效地估计第二基本形式的发展方程中的reaction项，分析解的存在性，收敛性，从而得到子流形的拓扑性质。这将是G. Pipoli 与C. Sinestrari关于复射影空间中闭子流形的相应工作的推广。另外，我们还将应用平均曲率流研究所有紧致型不可约的Hermitian对称空间及其子流形的辛微分同胚的形变，尤其是用有界对称域的方法统一处理U(n + m)/U(n) × U(m), n, m ≥ 1, SO(n + 2)/SO(n) × SO(2), n ≥ 3的情形。 这是 Guangcun Lu 与 Bang Xiao 在这方面工作的补充。