曲率流理论及其应用研究

本项目致力于曲率流理论及其应用的研究，以（带边值条件或不带边值条件的）逆曲率流、 Kähler-Ricci流作为主要的考虑对象，在不同的预设条件下，分析相关流的收敛性、渐近行为，同时，试图构造流下的单调量，以便得到一些有用的几何不等式（例如，Minkowski型不等式、Willmore型不等式、谱不等式、等周不等式等等）。.在已有研究成果的基础上，围绕上述研究内容，我们提出了具体问题，并且同时给出了可行的研究方案，这为项目的顺利展开奠定了基础。此外，不难发现，项目的主要研究内容为当前微分几何研究中的热点问题。

This project devotes to the study of the theory of curvature flows and its applications. We mainly consider inverse curvature flows (with or without boundary conditions), the Kähler-Ricci flow, and, under different settings, analyze the convergence and the asymptotical behavior of the flows considered. Besides, we also try to construct monotone quantities under the flows such that some useful geometric inequalities (e.g., Minkowski-type inequalities, Willmore-type inequalities, spectral inequalities, isoperimetric inequalities, etc) can be obtained..Based on our existing results, we propose specified problems about the above research contents, and meanwhile, we also give feasible approaches to those problems, which lay the foundation for carrying out the project well. Besides, it is not difficult to find that the main research contents of this project are hot topics in the current research of Differential Geometry.