Ricci流的研究是当今几何分析的主流方向.由于PERELMAN和HAMILTON在Ricci流问题上的杰出工作,人们认识到,通过研究流形上RICCI流的整体形态,也就是用黎曼几何和偏微分方程作为主要工具来研究几何流,可以搞清楚流形的拓扑.研究RICCI流的整体行为是一个基本问题.根据HAMILTON的爆破分析,RICCI流的整体行为分类成3种奇点.对于第一类奇点,PERELMAN做出了杰出的贡献.根据HAMILTON的猜想,经过合理重整化,RICCI流的可以收敛到RICCI孤立子.我们将进一步研究这个猜想,特别要研究非紧完备流形上的对应于第三类的RICCI流,改良PERELMAN的ENTROPY估计,约化距离和约化体积,和HAMILTON 的HARNACK估计来发展RICCI流理论.我们也将研究相关的几何流,比如YAMABE流问题.

The study of Ricci flow is one of mainstreams in Geometric analysis. Because of the outsanding works of G.Perelman and R.Hamilton,researchers have realized that, based on the knowledge of Riemannian geometry and PDE, people can know the topology of the manifolds via the study of the global behavior of Geometric flows auch as Ricci flow. The study of global Ricci flow is of fundamental importance.According to the blow-up analysis of Hamilton, the flow can be divided into three types.For type I flow,Perelman has made a remarkable contribution. According to Hamilton's conjecture, after a renomalization of the Ricci flow, the limit of the flow is a Ricci soliton. We shall study this conjecture, in particular, we shall study the type III Ricci flow on non-compact complete Riemannian manifolds. We shall further develop the Ricci flow theory by more refined study of Perelman's entropy functionals, reduced distance and volume, and Hamilton's Harnack type inequalities.We shall also study related geometric flows such as Yamabe flows.