Singularity Studies in the Ricci Flow and Kaehler-Ricci Flow

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The Ricci flow, and the field of geometric flows in general, has seen tremendous progress and yielded important applications to geometry, topology, and nonlinear analysis. The fundamental works of R. Hamilton and recent breakthrough by Perelman on the Ricci flow have led to the spectacular applications to geometrization of 3-manifolds, including the Hamilton-Perelman proof of the Poincaré conjecture.In this project, the PI will investigate several important problems related to the formation of singularities in the Ricci flow and the Kahler-Ricci flow which are of great interest in geometry, complex analysis, and nonlinear partial differential equations. They include studying the geometry, such as volume growths and curvature decay rates, and the classification of complete noncompact gradient shrinking Ricci solitons; the geometry of steady Ricci solitons with positive curvature, in particular the uniqueness question in 3 dimensions; asymptotic behavior of solutions to the Kahler-Ricci flow on compact Kahler manifolds with positive first Chern class.Progress on these issues would lead to profound new understandings in geometry and nonlinear analysis.The Ricci flow is an important type of geometric flows, which have profound importance and applications in science and geometry.Examples of applications of other include the motion of a surface by its mean curvature, the flow of gas in a porous mechanism, the motion of a liquid crystal, the diffusion of oil in shale, the reproduction of sparse species, and image sharpening.