FRG: Collaborative Research: Heat Equations and Geometric Flows in Riemannian and Kaehler Geometry

我们建议进一步发展有关佩雷尔曼的工作， 几何化猜想，利用Ricci流穿越时空 公式，新的梯度估计和单调量及其 几何应用，并应用新的技术和应用 基于这些方法和估计的想法，目的是 促进对以下相互关联的主题的理解： 1.紧和非紧Kahler流形的一致化， 结合Kahler-Ricci流和全纯 职能/科。 2.奇异性分析、弱解的公式化和流过去 几何发展方程的奇性和 Ricci流和平均曲率流。 3.调和/全纯函数、函数论、谱的研究 和完备黎曼/卡勒流形的几何。 4.上的Einstein和其他正则黎曼度量的存在性 流形 几何演化方程是研究这一问题的重要工具 流形的整体几何和拓扑Perelman最近的工作 关于Ricci流的汉密尔顿程序及其在 庞加莱和几何化几何的可能解决方案提供了一个及时的 这是一个很有希望的机会， 在几何分析和相关领域。该项目的成果 应该导致弦/对偶理论和 重正化群流、Ricci流、平均曲率流等 几何演化方程，并可能提高理解的 宇宙在大尺度上的均匀性，以及其他领域， 科学该项目将提高对几何分析的理解， 线性和非线性偏微分方程，代数几何和 数学物理

We propose to further developments related to Perelman's work on the Geometrization Conjecture, using Ricci flow through the space-time formulation, new gradient estimates and monotone quantities and their geometric applications, and to apply new techniques and applications based on the ideas of these methods and estimates with the aim of furthering the understanding of the following interconnected topics: 1. Uniformization of compact and noncompact Kahler manifolds, combining Kahler-Ricci flow and the study of holomorphic functions/sections. 2. Analysis of singularities, formulation of weak solutions, and flow past singularities for geometric evolution equations and the duality between the Ricci flow and the mean curvature flow. 3. Study of harmonic/holomorphic functions, function theory, spectrum and the geometry of complete Riemannian/Kahler manifolds. 4. Existence of Einstein and other canonical Riemannian metrics on manifolds. Geometric evolution equations are powerful and central tools in the study of the global geometry and topology of manifolds. Recent work of Perelman on Hamilton's program for Ricci flow and its applications towards a possible solution to the Poincare and geometrization conjectures provides a timely and promising opportunity for a group effort on significant advancements in geometric analysis and related areas. The results from the project should lead to new advances in and connections between string/duality theory and renormalization group flow, Ricci flow, mean curvature flow and other geometric evolution equations, and may enhance the understanding of the homogeneity of the universe at large scales, as well as other areas in science. The project will enhance the understanding of geometric analysis, linear and nonlinear partial differential equations, algebraic geometry and mathematical physics.