本项目拟研究黎曼流形上的特殊几何结构及相关分类. 主要的研究内容如下: 1) 研究Critical point方程, V-Static方程以及Miao-Tian Critical metric方程的分类; 2) 研究二次曲率泛函临界点方程的分类; 3) 研究具有特殊几何结构 (例如, 具有平坦的Bach张量, 调和的黎曼曲率张量, 平行的Cotton张量) 的(子)流形的分类; 4) 研究Self-shrinkers, 具有Ricci Soliton结构或quasi-Einstein结构的(子)流形的分类. 另外, 考虑Bakry-Emery Ricci曲率有关的几何量, 研究Witten Laplacian的特征值估计及相关分类. 主要方法是利用流形的特殊几何结构, 开发张量之间的精确关系式, 得到分类与刚性刻画.

The aim of this project is to study special geometric structures and related classifications on Riemannian manifolds. Main contents are the following: 1) classifications of Critical point equations, V-Static equations and Miao-Tian Critical metric equations; 2) classifications of critical equations of the quadratic curvature functionals; 3) classifications of Riemannian (submanifolds) manifolds with special geometrical structures (for example, manifolds with Bach-flat metric, with harmonic Riemannian curvature tensor, or with prallel Cotton tensor) ; 4) classifications of (submanifolds) manifolds with the structures of Self-shrinkers, Ricci solitons or quasi-Einstein manifolds. Moreover, we study manifolds with respect to the Bakry-Emery Ricci curvature, eigenvalue estimates and related classifications of Witten Laplacian. By using special geometric structures and exploiting the precise estimates with respect to the tensors, we will give some classifications and rigidity theorems.