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步，并在几何、拓扑和非线性分析方面产生了重要的应用。汉密尔顿的基本工作和佩雷尔曼最近在里奇流上的突破导致了3流形几何化的惊人应用，包括汉密尔顿-佩雷尔曼对庞卡罗猜想的证明。在这个项目中，PI将研究与Ricci流和Kahler-Ricci流奇点形成有关的几个重要问题，这些问题在几何、复杂分析和非线性偏微分方程中都有很大的兴趣。它们包括研究几何，如体积增长和曲率衰减率，以及完全非紧化梯度收缩里奇孤子的分类；具有正曲率的稳定Ricci孤子的几何，特别是三维中的唯一性问题；具有正第一类的紧Kahler流形上Kahler- ricci流解的渐近行为。在这些问题上的进展将导致对几何和非线性分析的深刻的新认识。利玛窦流是一种重要的几何流类型，在科学和几何中具有深远的意义和应用。其他应用的例子包括平均曲率的表面运动，多孔机制中的气体流动，液晶的运动，页岩中油的扩散，稀疏物种的复制和图像锐化。

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.