Singularity Studies in the Ricci Flow and Kaehler-Ricci Flow

里奇流，以及一般的几何流领域，已经取得了巨大的进展，并在几何、拓扑和非线性分析中产生了重要的应用。介绍了R.汉密尔顿和Perelman最近在Ricci流上的突破导致了三维流形几何化的惊人应用，包括Poincaré猜想的Hamilton-Perelman证明。在这个项目中，PI将研究与Ricci流和Kahler-Ricci流中奇点形成有关的几个重要问题，这些问题在几何学，复分析，非线性偏微分方程其中包括研究完全非紧梯度收缩Ricci孤子的体积增长、曲率衰减率等几何性质和分类，正曲率稳定Ricci孤子的几何性质，特别是三维唯一性问题; Kahler方程解的渐近行为Ricci流是一个具有正第一Chern类的紧致Kahler流形上的Ricci流，这些问题的研究将对几何学和非线性分析产生新的深刻的认识，Ricci流是一个具有正第一Chern类的紧致Kahler流形上的Ricci流。一种重要的几何流动类型，在科学和几何学中有着深远的重要性和应用。其他应用的例子包括表面平均曲率的运动、多孔机制中的气体流动、液晶的运动、页岩中石油的扩散、稀疏物种的再现和图像锐化。

The Ricci flow, and the field of geometric flows in general, has seen tremendous progress and yielded important applications to geometry, topology, and nonlinear analysis. The fundamental works of R. Hamilton and recent breakthrough by Perelman on the Ricci flow have led to the spectacular applications to geometrization of 3-manifolds, including the Hamilton-Perelman proof of the Poincaré conjecture.In this project, the PI will investigate several important problems related to the formation of singularities in the Ricci flow and the Kahler-Ricci flow which are of great interest in geometry, complex analysis, and nonlinear partial differential equations. They include studying the geometry, such as volume growths and curvature decay rates, and the classification of complete noncompact gradient shrinking Ricci solitons; the geometry of steady Ricci solitons with positive curvature, in particular the uniqueness question in 3 dimensions; asymptotic behavior of solutions to the Kahler-Ricci flow on compact Kahler manifolds with positive first Chern class.Progress on these issues would lead to profound new understandings in geometry and nonlinear analysis.The Ricci flow is an important type of geometric flows, which have profound importance and applications in science and geometry.Examples of applications of other 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, the reproduction of sparse species, and image sharpening.