Analytic and Geometric Aspects of Ricci Flow

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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.