Ricci Flow, Kaehler-Ricci Flow and Applications

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AbstractAward: DMS-0506084Principal Investigator: Huai-Dong CaoThe Ricci flow, introduced by Richard Hamilton, has become one ofthe most powerful tools in geometric analysis. In the past twentyyears or so, Hamilton has proved many remarkable theorems in theRicci flow and developed a remarkable program to approach thePoincare conjecture and Thurston's geometrization conjectureusing the Ricci flow. More recently, Perelman has madeastounding breakthrough in the Ricci flow with the proof of alocal injectivity radius estimate valid for all dimensions andused it to study the geometrization of three-manifolds. Inaddition to the applications of the Ricci flow tothree-manifolds, many exciting possibilities remain. In thisproposal, we propose to investigate several important problems inthe Ricci flow and the Kaehler-Ricci flow which are of greatinterest in geometry, topology, nonlinear partial differentialequations and complex analysis. They include studyingstability/instability of Einstein metrics of positive scalarcurvature (and more generally of shrinking Ricci solitons),constructing new Ricci solitons, seeking new Einstein metrics viathe Ricci flow, aspects of geometrization of 4-manifolds,studying the asymptotic behavior of solutions to theKaehler-Ricci flow on compact Kaehler manifolds with positivefirst Chern class, and the uniformization of complete noncompactKaehler manifolds of positive curvature.The Ricci flow is an important type of geometric flows (orgeometric evolution equations) which have profound importance andapplications in science and geometry. Examples of applications ofother include the motion of a surface by its mean curvature, theflow of gas in a porous mechanism, the motion of a liquidcrystal, the diffusion of oil in shale, the reproduction ofsparse species, and image sharpening.