极值子流形的几何刚性问题

In this thesis, we study the geometric rigidity of Willmore submanifolds and extremal submanifolds in a sphere，including the rigidity problem under a pinching condition for Willmore submanifolds and extremal submanifolds in a sphere，the gap phenomenon for the eigenvalues of Schrodinger operator on Willmore submanifolds and extremal submanifolds, and the spectrum of Laplace operator on extremal hypersurfaces.

本项目主要研究了Willmore子流形和极值子流形的几何刚性问题。包括球面中Willmore子流形与极值子流形在拼挤条件下的刚性问题。另外还研究这两类子流形上薛定谔算子特征值的刚性问题，以及极值子流形谱与几何的问题。

设球面中的极值超曲面M与Willmore环W具有相同的平均曲率。若M与W是等谱的，则M必定是Willmore环W。还证明了球面中极值子流形的一个刚性定理。