On the Kaehler-Ricci Flow and Related Problems

关于凯勒-里奇流及相关问题

• 批准号: 0206847
• 负责人: Huai-Dong Cao
• 金额: $ 10.7万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0206847&HistoricalAwards=false
• 关键词: Kaehler; Ricci; Flow; Related; Problems

ABSTRACT DMS - 0206847.Abstract of the ProjectThis project is centered around the study of the Ricci flow on complete,non-compact Kaehler manifolds of nonnegative holomorphic bisectionalcurvature. The main goal is to seek a complete understanding of singularbehavior of the maximal solutions to the Kaehler-Ricci flow withnonnegative curvature, in particular the geometry, such as the rates of curvature decay and volume growth of geodesic balls, of complete Kaehler-Ricci solitons,which turn out to be the models for Type II singularities and can be considered as anatural extension of Calabi-Yau metrics on non-compact complexmanifolds. Progress on the project will lead to new understanding of geometry, analysis, and complex structure and could have important application to solving awell-known conjecture in complex geometry. The research will be basedon the effort of extending the dimension reduction method of Hamiltonfor the Ricci flow in the Riemannian case to the Kaehler case where onehas weaker curvature assumption. The proposed project deals with singular behavior of the Kaehler-Ricciflow, or Parabolic-Einstein equations, and studies uniformization typeproblems for complete Kaehler manifolds. The Kaehler-Ricci flow is animportant type of geometric evolution equations, which have profoundimportance and applications in science and geometry. Some of the examples 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, thereproduction of sparse species, and image sharpening. The knowledgegained in this project would not only have significant implications toour understanding of complex geometry which in turn could be very usefulto the study of mathematical physics, but also may lead to newunderstanding of singularity formations in other geometric evolutions.------------------------------------------------------------------------

DMS -0206847.项目摘要本项目主要研究具有非负全纯双曲曲率的完备非紧Kaehler流形上的Ricci流。本文的主要目的是对具有非负曲率的Kaehler-Ricci流的极大解的奇异性态，特别是完全Kaehler-Ricci孤子的几何性质，如曲率衰减率和测地球体积增长率，有一个完整的理解，它是第二类奇异性的模型，可以被认为是Calabi-Yau度量在非紧复流形上的自然推广。该项目的进展将导致对几何、分析和复杂结构的新理解，并可能对解决复杂几何中的一个著名猜想有重要的应用。 本文的研究工作是将黎曼情形下Ricci流的Hamilton降维方法推广到具有较弱曲率假设的Kaehler情形。本项目主要研究Kaehler-Ricciflow方程或抛物-爱因斯坦方程的奇异性，并研究完备Kaehler流形的一致化问题。Kaehler-Ricci流是一类重要的几何发展方程，在科学和几何中有着重要的意义和应用。其中一些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，页岩中石油的扩散，稀疏物种的再生和图像锐化。在这个项目中获得的知识不仅对我们理解复杂几何有重要意义，这反过来又对数学物理的研究非常有用，而且可能导致对其他几何演化中奇点形成的新理解。------------