Analytic and Geometric Aspects of Ricci Flow

项目编号：dms -0505507项目负责人：Bennett chow本项目旨在研究Ricci流和相关几何演化方程领域的几何和解析问题。在奇点分析的激励下，获得对古代解的进一步理解是很有意义的。我们利用新的哈纳克不等式、熵估计和在第2维特别有用的Aleksandrov反射方法，研究了获得更详细的第2维和第3维分类的问题。特别是Hamilton和perelman熵估计、li - you -Hamilton型哈纳克估计、梯度估计、时空几何、l-函数、线性化里奇流等技术，可以推广并有助于解决这些问题。研究了一类奇异点的存在性问题。交叉曲率流对于理解3流形上具有负截面曲率的度量空间是有用的。我们建议研究它的长期行为。说明性书籍项目将进一步理解几何分析和几何演化方程在更大的数学和科学社区。里奇流是研究流形特别是低维流形的分析、几何和拓扑的重要工具。它与其它几何演化方程密切相关。佩雷尔曼最近对里奇流的Hamilton程序及其对庞加莱猜想和几何化猜想的可能解的应用的研究，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里奇流和其他几何演化方程的问题。这将增加我们对流形的分析和几何以及相关领域的理解，如拓扑，偏微分方程和数学物理。

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.