On the Kaehler-Ricci Flow and Related Problems

摘要DMS - 0206847。项目摘要本项目主要研究非负全纯等分曲率的完全非紧化Kaehler流形上的Ricci流。主要目标是寻求对非负曲率Kaehler-Ricci流的极大解的奇点行为的完全理解，特别是完全Kaehler-Ricci孤子的几何形状，如曲率衰减率和测地球的体积增长，这被证明是II型奇点的模型，可以被认为是Calabi-Yau度量在非紧复流形上的自然扩展。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众所周知的猜想具有重要应用。将黎曼情况下Ricci流的hamilton降维方法推广到曲率假设较弱的Kaehler情况。提出的项目处理Kaehler- ricciflow或抛物-爱因斯坦方程的奇异行为，并研究完全Kaehler流形的均匀化型问题。Kaehler-Ricci流是一类重要的几何演化方程，在科学和几何领域具有重要的意义和应用。其中的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，页岩中油的扩散，稀疏物种的产生以及图像锐化。在这个项目中获得的知识不仅对我们对复杂几何的理解有重要的意义，反过来对数学物理的研究非常有用，而且还可能导致对其他几何进化中的奇点形成的新理解。------------------------------------------------------------------------

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.------------------------------------------------------------------------