等参超曲面及其相关问题

Some geometric problems of isoparametric hypersurfaces in the sphere, especially the geometric properties of the corresponding focal submanifolds are studied in this project. The isoparemetric hypersurfaces are one of the most important objects in the submanifold geometry, in which to study the geometric properties of isoparametric hypersurfaces and their focal submanifolds has the profound significance. In this project, by the further study on Lie groups and Lie algebras, we should consider the first eigenvalues of the focal submanifolds in the unsolvable cases settled by Tang and Yan. Moreover, we should study the problems whether the focal submanifolds are Willmore or Einstein in the last unsolvable cases settled by Qian, Tang and Yan, especially the case g=6. The main challenges of this project are how to use the Lie groups and Lie algebras and the method provided by Tang and Yan to study the related topics of focal submanifolds.

本项目主要研究球面上的等参超曲面及其相关的问题，特别是其所对应的焦流形的几何性质。等参超曲面是子流形几何中重要的一类研究对象，研究等参超曲面及其焦流形的性质是子流形几何的重要研究课题之一。本项目拟通过对李群与李代数的更深入的讨论，我们将对球面上焦流形（剩余未解决情形）第一特征值问题的进行研究；同时，拟通过对焦流形几何性质进一步的研究，对剩余未解决情形特别是g=6情形时，指出它们是否都是Willmore子流形以及Einstein 流形。主要挑战是如何通过李群与李代数的计算以及唐和彦的方法，完成对焦流形的相关几何问题的研究。

本项目主要研究球面上的等参超曲面及其相关的问题，特别是其所对应的焦流形的几何性质。通过对焦流形的更深入的讨论，给出了焦流形（剩余未解决情形）的第一特征值问题的估计；同时，通过对焦流形几何性质进一步的研究，对剩余未解决情形特别是g=6情形时，证明了它们都是Willmore子流形。