Analytic and Geometric Aspects of Ricci Flow

AbstractAward：DMS-0505507主要研究者：班尼特周该项目旨在研究里奇流和相关几何演化方程领域的几何和分析问题。受奇异性分析的启发，对古代解有了进一步的认识是有意义的。我们研究了利用新的Harnack不等式、熵估计和在2维中特别有用的Aleksandrov反射方法在2维和3维中获得更详细分类的问题。特别是汉密尔顿和Perelman熵估计、Li-Yau-汉密尔顿型Harnack估计、梯度估计、时空几何、l-函数、线性化Ricci流等，都是解决这些问题的可推广和有用的技术。 考虑了第二类奇异性的存在性问题。交叉曲率流对于理解三维流形上具有负截面曲率的度量空间是有用的。我们打算研究它的长期行为。这本书将进一步加深数学和科学界对几何分析和几何演化方程的理解。里奇流是研究流形的分析、几何和拓扑的重要工具，特别是在低维情况下。它与其它几何发展方程有着密切的联系。Perelman关于Ricci流的汉密尔顿程序及其在Poincare方程和几何化方程的可能解中的应用的最新工作，产生了许多新的思想和技术，这些思想和技术可能适用于解决Ricci流和其它几何发展方程的问题。这将增加我们对流形的分析和几何以及相关领域如拓扑学、偏微分方程和数学物理的理解。

AbstractAward: DMS-0505507Principal Investigator: Bennett ChowThe project is aimed at the study geometric and analytic problemsin the field of Ricci flow and related geometric evolutionequations. Motivated by singularity analysis, it is of interestto obtain further understanding of ancient solutions. We studythe problem of obtaining a more detailed classification indimensions 2 and 3 using new Harnack inequalities, entropyestimates and the method of Aleksandrov reflection which isparticularly useful in dimension 2. In particular Hamilton andPerelman entropy estimates, Li-Yau-Hamilton type Harnackestimates, gradient estimates, space-time geometry, l-function,linearized Ricci flow are techniques which may be generalizableand useful to solve these problems. The existence problem forType II singularities is considered. The cross curvature flowshould be useful in understanding the space of metrics withnegative sectional curvature on a 3-manifold. We propose toinvestigate its long time behavior. The expository book projectswill further the understanding of geometric analysis andgeometric evolution equations in the greater mathematical andscientific communities.Ricci flow is an important tool in the study of the analysis,geometry and topology of manifolds, especially in lowdimensions. It is intimately related to other geometric evolutionequations. The recent work of Perelman on Hamilton's program forRicci flow and its applications towards a possible solution tothe Poincare and geometrization conjectures has yielded aplethora of new ideas and techniques which may be applicable tosolve problems in Ricci flow and other geometric evolutionequations. This will increase our understanding of the analysisand geometry of manifolds and related fields such as topology,partial differential equations, and mathematical physics.