Ricci Flow, Kaehler-Ricci Flow and Applications

由Richard Hamilton提出的Ricci流已成为几何分析中最强大的工具之一。在过去的二十多年里，汉密尔顿证明了里奇流中的许多重要定理，并开发了一个了不起的程序来利用里奇流来接近庞加莱猜想和瑟斯顿的几何化猜想。最近，Perelman在Ricci流中取得了惊人的突破，证明了局部注入半径估计在所有维度上都是有效的，并将其用于研究三流形的几何化。除了将里奇流应用于三流形之外，还有许多令人兴奋的可能性。在本提案中，我们建议研究Ricci流和Kaehler-Ricci流在几何，拓扑，非线性偏微分方程和复分析方面的几个重要问题。它们包括研究正标量曲率的爱因斯坦度量的稳定性/不稳定性（以及更一般的收缩Ricci孤子），构造新的Ricci孤子，通过Ricci流寻求新的爱因斯坦度量，4-流形的几何化方面，研究Kaehler-Ricci流在具有正第一chen类的紧化Kaehler流形上解的渐近行为，以及正曲率的完全非紧化Kaehler流形的均匀化。利玛窦流是一种重要的几何流（或几何演化方程），在科学和几何中具有深远的重要性和应用。其他应用的例子包括表面的平均曲率运动，多孔机制中的气体流动，液晶的运动，页岩中油的扩散，稀疏物种的再现和图像锐化。

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.