Analytic and Geometric Aspects of Ricci Flow

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Abstract Award: DMS-0505507 Principal Investigator: Bennett Chow The project is aimed at the study geometric and analytic problems in the field of Ricci flow and related geometric evolution equations. Motivated by singularity analysis, it is of interest to obtain further understanding of ancient solutions. We study the problem of obtaining a more detailed classification in dimensions 2 and 3 using new Harnack inequalities, entropy estimates and the method of Aleksandrov reflection which is particularly useful in dimension 2. In particular Hamilton and Perelman entropy estimates, Li-Yau-Hamilton type Harnack estimates, gradient estimates, space-time geometry, l-function, linearized Ricci flow are techniques which may be generalizable and useful to solve these problems. The existence problem for Type II singularities is considered. The cross curvature flow should be useful in understanding the space of metrics with negative sectional curvature on a 3-manifold. We propose to investigate its long time behavior. The expository book projects will further the understanding of geometric analysis and geometric evolution equations in the greater mathematical and scientific communities. Ricci flow is an important tool in the study of the analysis, geometry and topology of manifolds, especially in low dimensions. It is intimately related to other geometric evolution equations. The recent work of Perelman on Hamilton's program for Ricci flow and its applications towards a possible solution to the Poincare and geometrization conjectures has yielded a plethora of new ideas and techniques which may be applicable to solve problems in Ricci flow and other geometric evolution equations. This will increase our understanding of the analysis and geometry of manifolds and related fields such as topology, partial differential equations, and mathematical physics.