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

我们建议进一步发展Perelman关于几何化猜想的工作，利用Ricci流通过时空公式、新的梯度估计和单调量及其几何应用，并基于这些方法和估计的思想应用新的技术和应用，目的是加深对以下相互关联的主题的理解：1.紧致和非紧致Kahler流形的统一，结合Kahler-Ricci流和全纯函数/截面的研究。2.几何发展方程的奇性分析、弱解的表示、通过奇点的流动以及Ricci流和平均曲率流之间的对偶性。3.研究调和/全纯函数、函数论、谱和完备黎曼/Kahler流形的几何。4.流形上爱因斯坦和其他正则黎曼度量的存在性。几何演化方程是研究流形全局几何和拓扑的强大而核心的工具。佩雷尔曼最近在哈密尔顿的Ricci流程序及其应用方面的工作，为Poincare和几何化猜想的可能解决方案提供了一个及时和有希望的机会，为小组在几何分析和相关领域的重大进展提供了一个机会。该项目的结果将导致弦/对偶理论与重整化群流、Ricci流、平均曲率流和其他几何演化方程之间的新进展和联系，并可能加强对大尺度宇宙同质性的理解，以及科学上的其他领域。该项目将加强对几何分析、线性和非线性偏微分方程式、代数几何和数学物理的理解。

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.