Laplace和体积比较定理，及由此发展出的Cheeger-Colding-Tian-Naber结构理论是研究黎曼流形几何和拓扑性质的重要工具。申请人在这方面的主要成果有：1.证明了Bakry-Émery Ricci曲率条件下的比较定理，推广了大部分的结构理论，包括Cheeger-Naber著名的余维4定理；2.将Anderson调和半径估计的正则性减弱了整整一阶，使该结果在20多年后取得进一步突破；3.三十多年来首次不假设Ricci曲率有下界，证明了仅依赖于Ricci曲率积分界的Li-Yau不等式，并研究了它们与Laplace比较定理的关系。在此基础上，本课题将围绕以下问题展开：1.在Bakry-Émery Ricci曲率条件下进一步改进比较定理并推广结构理论；2.改进Ricci曲率积分条件下的Li-Yau不等式，证明Laplace比较定理；3.探索更一般曲率条件下的比较定理和结构理论。

The Cheeger-Colding-Tian-Naber structure theory was developed from Laplace and Volume Comparison Theorems. Comparison theorems and the structure theory are essential tools in the study of the geometric and topological properties of Riemannian manifolds. Main achievements of the applicant include: 1. established comparison theorems under Bakry-Émery Ricci curvature condition, and generalized the majority of the structure theory which includes the famous codimension 4 theorem of Cheeger and Naber; 2. proved that the regularity of Anderson's harmonic radius estimate can be significantly reduced by exactly 1 order, which is a further improvement on this result after more than 20 years；3. for more than 30 years, all Li-Yau inequalities on Riemannian manifolds are established under the assumption that the Ricci curvature has certain point-wise lower bound. We are the first ones who discovered Li-Yau inequalities based on integral bounds of the Ricci curvature, which is a much weaker condition. Moreover, the relation between Li-Yau inequalities and Laplace comparison theorem was investigated. Based on previous works, this project will concentrate on the following problems: 1. further improve the comparison theorems and generalize the structure theory under Bakry-Émery Ricci curvature condition; 2. refine Li-Yau inequalities and prove Laplace comparison theorem under integral Ricci curvature conditions; 3. explore volume comparison theorem and the structure theory under more general curvature conditions.