Analytic and Geometric Aspects of Ricci Flow

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ABSTRACT DMS - 0203926.The objective of this project is to study analytic and geometric aspects of Hamilton's Ricci flow of Riemannian metrics, and related topics. In particular, the principal investigator proposes to investigate injectivity radius estimates for solutions to the Ricci flow. The PI proposes to studyLi-Yau-Hamilton (Harnack) inequalities for both the Ricci flow and the linearized Ricci flow. The PI also proposes to study collapsing sequences of solutions to the Ricci flow and the related compactness theory. Finally, the PI proposes to study estimates for the linearized Ricci flow andtheir applications. The PI will focus on dimension 3, where there are the most tools available and the problems posed have the most topological applications.The Ricci flow is an evolution equation which deforms geometric structures on surfaces and manifolds both improving and smoothing out the structures. It has been at the cutting edge of the recent rapid progress in geometric evolution equations and has greatly influenced the study of the mean curvatureflow and many other geometric evolution equations. The work on understanding the singularities which develop under the flow has led to the development of new and powerful analytic and geometric tools, which have found applications in thestudy of many geometric evolution equations. Many phenomena aremodeled by geometric evolution equations, such as heat transfer, the evolution of interfaces between molten and solid metal, and the interfaces between ice and water. Geometric evolution equations are also used to smooth out images and have applications in computer vision.