Some Analytic and Geometric Problems of the Ricci Flow

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The project is aimed at the study of geometric evolution equations and related topics, especially the Ricci flow. We study Li-Yau-Hamilton type Harnack inequalities, their relations with Perelman?s entropy functional, and their applications to geometric flows. Motivated by Hamilton?s work on singularities analysis, we study properties of singularities and ancient solutions of the Ricci flow on 4-manifolds with symmetry. The monotonicity of various geometric operators under the Ricci flow is considered. We propose to investigate geometric flows on locally homogeneous manifolds. This will help us to understand the long time behavior of geometric flows better. The Ricci flow has been a center of attention of differential geometry and topology. Recent work of Perelman on Hamilton?s program of the Ricci flow gives a solution to the long-standing Poincare conjecture and also a possible solution to the geometrization conjecture. The proposed project will improve our understanding of the Ricci flow program and as well as other geometric evolution equations, such as mean curvature flow, Gauss curvature flow, cross curvature flow and renormalization group flow. The project will also enhance our understanding of other fields in differential geometry, parabolic PDE and mathematical physics. The education component will help graduate students or researchers in other fields to understand the Ricci flow program better.