黎曼流形与子流形上的特征值及相关问题

黎曼流形上的特征值问题是微分几何和几何分析领域中的一个重要课题，因为特征值对于刻画流形本身的几何性质提供了非常重要的信息。另一方面，子流形上的一些椭圆算子的特征值的信息反映了子流形本身的稳定性、刚性等, 而稳定性结果与刚性定理是子流形几何研究中的一个重要方面。目前，仍有很多特征值问题的最优估计是不清楚的。 综上可知，特征值问题及相关问题的研究是一个非常有意义且重要的研究课题。本项目拟在已有工作基础上，重点对黎曼流形和子流形上的特征值及其相关问题展开研究。研究内容分三部分：（1）研究完备黎曼流形的有界区域上一些特征值问题的特征值估计；（2）研究子流形上一些椭圆算子的特征值估计；（3）研究子流形的刚性。譬如：欧氏空间下平均曲率流中self-shrinkers的刚性和分类问题等。

The eigenvalue problems are very interesting and important problems in the fields of differential geometry and geometry analysis because one can obtain some important information from the estimates of the eigenvalues. On the other hand, one can get some interesting stability results and rigidity theorems from the estimates of some elliptic operators on submanifolds. And the stability results and rigidity theorems are nice results in the field of submanifolds. At present, one can not obtain the optimal estimates of eigenvalues for many eigenvalue problems. From the above arguments, we know that it is very interesting and important to study eigenvalue problems and related problems. Based on previous work, in this project, we are going to investigate eigenvalue problems on Riemannian manifolds, submanifolds and related problems. Specifically, we will try to study: (1) Eigenvalue estimates on bounded domains in complete Riemannian manifolds; (2) Eigenvalue estimates of elliptic operators on submanifolds; (3) The rigidity results of submanifolds, for example: the rigidity and classification of self-shrinkers.