Ricci Flow, Kaehler-Ricci Flow and Applications

Ricci 流、Kaehler-Ricci 流和应用

• 批准号: 0506084
• 负责人: Huai-Dong Cao
• 金额: $ 10.8万
• 依托单位: Lehigh University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0506084&HistoricalAwards=false
• 关键词: Ricci; Flow; Kaehler; Applications

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.

摘要奖：DMS-0506084主要研究员：曹怀东理查德·汉密尔顿提出的Ricci流已成为几何分析中最强大的工具之一。在过去的二十年左右的时间里，哈密尔顿证明了Ricci流中的许多重要定理，并开发了一个非凡的程序来利用Ricci流来逼近Poincare猜想和瑟斯顿几何猜想。最近，Perelman在Ricci流中取得了惊人的突破，证明了局部内射性半径估计对所有维度都有效，并将其用于研究三维流形的几何化。除了Ricci流在三维流形上的应用外，还有许多令人兴奋的可能性。在这个方案中，我们建议研究Ricci流和Kaehler-Ricci流中的几个重要问题，这两个问题在几何、拓扑、非线性偏微分方程组和复分析中都有很大的兴趣。它们包括研究正标量曲率的爱因斯坦度规(以及更一般的收缩Ricci孤子)的稳定性/不稳定性，构造新的Ricci孤子，通过Ricci流寻找新的爱因斯坦度规，4-流形的几何化方面，研究具有正第一类的紧致Kaehler流形上Kaehler-Ricci流解的渐近行为，以及完全非紧正曲率Kaehler流形的均匀化。Ricci流是一类重要的几何流(或几何演化方程)，在科学和几何中具有重要的意义和应用。其他应用的例子包括表面的平均曲率运动、气体在多孔机构中的流动、液晶的运动、石油在页岩中的扩散、稀疏物种的繁殖和图像锐化。