Singularity Studies in the Ricci Flow and Kaehler-Ricci Flow

Ricci流，以及整个几何流领域，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。R·哈密尔顿的基础工作和最近Perelman关于Ricci流的突破导致了三维流形几何化的壮观应用，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流的奇点形成有关的几个重要问题，这两个问题在几何、复分析和非线性偏微分方程组中都是非常感兴趣的。它们包括研究几何，如体积增长和曲率衰减率，完全非紧梯度收缩Ricci孤子的分类，定常正曲率Ricci孤子的几何，特别是三维唯一性问题；具有正第一类的紧致Kahler流形上Kahler-Ricci流的解的渐近性.在这些问题上的进展将在几何和非线性分析方面产生深刻的新认识.Ricci流是几何流的一种重要类型，在科学和几何中具有深远的重要性和应用.其他应用的例子包括：表面的平均曲率运动，气体在多孔机制中的流动，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.

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.