Nonlinear geometric evolution equations

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Abstract - DMS - 0805143 There are two classes of problems on which this project is focused. The first is about the geometric and analytic properties of the Ricci flow equation and their applications to the study of geometry and the topology of manifolds. Secondly, there are a couple of nonlinear partial differential equations appeared in the applied science, to which some techniques and results obtained by the PI are related and can be applied. The first part of project consists of the classifications of self-similar solutions, namely gradient solitons, and the convexity type estimates for Ricci flow solution in high dimensions. Such results have far-reaching consequences in the applications of Ricci flow in the study of geometric-topological structure of the manifolds. The second theme of the project is about the sharp gradient estimates of Li-Yau-Hamilton type, related monotonicity formulae and their applications in geometric nonlinear PDEs. Their relations to physics, statistical mechanics in particular, shall be studied too. The aim is to discover a fundamental physical/geometric principle to unify various sharp estimates and monotonicity formulae. It will also provide the guideline for further discovery of the new monotonicity formulae in other geometric partial differential equations. The project is on the geometric and analytic properties of Ricci flow, one of the most important geometric partial differential equation in mathematics and physics. The completion of the project will enhance the current understanding of the singularity formation of the Ricci flow equation and its relation with several branches of applied science and theoretic physics. By giving lectures for general audience including high school students, writing several monographs and survey articles, the project contributes to the dissemination of enhancing the scientific understanding of mathematical community as well as the general publics.