黎曼流形上的特征值在几何与物理上都有非常重要的应用，它与流形的几何、拓扑等有着密切的联系。本项目旨在研究Witten Laplacian的特征值及与其相关的Ricci Soliton几何结构。主要研究内容如下：（1）利用流形的几何性质并构造新的实验函数，研究m维Bakry-Emery Ricci曲率条件下的Witten Laplacian特征值间隙估计以及第一特征值估计，并研究m维Bakry-Emery Ricci曲率条件下的流形的直径估计；（2）利用几何不变量之间的关系式，研究quasi-Einstein流形的刚性和几何拓扑分类；（3）研究复空间形式中具有Ricci Soliton结构或Yamabe Soliton结构的实超曲面的几何与拓扑分类。以上研究是当前微分几何的研究热点，也是物理学家非常关注的问题。

Eigenvalues on Riemannian manifolds play an important role in geometry and physics, which are closely related with geometrical properties and topological structures of manifolds. The aim of this project is to study eigenvalues of the Witten Laplacian and related geometrical structures of Ricci solitons. Main contents are the following: (1) Making use of geometrical properties and constructing new appropriate trial functions, we will study the gap of eigenvalues and the estimate on the first eigenvalue of Witten Laplacian under the assumption of the m-dimensional Bakry-Emery Ricci curvature. Furthermore, we will consider the estimate for the diameter of manifolds with respect to m-dimensional Bakry-Emery Ricci curvature; (2) Some rigidity results and classifications on geometry and topology of quasi-Einstein manifolds will be studied by virtue of hidden relations between geometric invariants; (3) We also study real hypersurfaces of a complex space form admitting Ricci soliton or Yamabe soliton. These problems are very hot and are paid close attention by physicists.