Parabolic differential equations and the geometry of manifolds

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The proposed research concerns several problems in the study of nonlinear parabolic PDE and their application to the geometry and topology of manifolds. One general area of concentration will be that of singularity formation in the Ricci flow, with an emphasis on the study of ancient solutions and Ricci solitons. A central theme in the theory developed by Hamilton, Perelman, and others, is that the local geometry of a developing singularity possesses a relatively rigid structure, modelled, in many important cases, upon these special types of solutions. It is thus desirable to have as detailed a knowledge as possible of the diversity of forms which they can assume.Additionally, the co-PI proposes to investigate Type-II singularity development and the question of unique continuation for the Ricci flow equation. Another general area of concentration will be curvature flows of hypersurfaces, with an emphasis on the application, interpretation, and further development of differential Harnack inequalities for these flows. One aim will be to refine such inequalities in the setting of evolving spacelike hypersurfaces in Minkowski space, with an eye toward the study of eternal solutions and translating solitons. In the case of the mean curvature flow, the latter objects are of interest to researchers in general relativity as natural foliations of Lorentzian spacetimes. In another direction, the co-PI proposes to pursue a potential connection from this setting to the cross-curvature flow, an intrinsic flow of potential use in the study of three-manifolds with negative curvature.The Ricci flow and other geometric evolution equations considered in this proposal are representatives of the "heat-flow" method in geometry, the techniques and objectives of which straddle the field's lively interface with topology, analysis, and mathematical physics.This method has proven effective in attacking certain cases of one of the most fundamental questions in mathematics, namely, which manifolds admit constant curvature or otherwise canonical geometries? Not only does this question have ramifications for physical models of the universe, but the development of tools attendant to the approach promise to pay continued dividends to the analysis of the many structurally similar nonlinear PDE which occur as models of diverse phenomena throughout the physical sciences.