Mean curvature flow and Ricci flow

该项目的目的是研究热方程的两个几何版本：表面的平均曲率的演变，以及弯曲空间的里奇流的演变。平均曲率流模拟了许多物理过程，这些过程涉及到一个不断变化的表面或界面。这是最有效的方法来减少表面的面积，并向最佳的发展。相应地，Ricci流使弯曲空间朝向最佳形状变形。虽然在这两种流动上已经获得了许多基本结果，但一个核心问题是在大多数相关情况下会形成奇点。PI的拟议研究将提供理解这些奇点并继续通过它们的方法。这将促进数学内外的许多新应用。拟议的研究是在平均曲率流和里奇流，重点是奇点的形成和技术，继续通过奇点的流动。在与布鲁斯克莱纳的联合工作中，PI将根据他们最近对平均凸流的估计，给出一个新的平均曲率流的构造。新的结构比以前的结构更基本，也更短。PI将应用平均曲率流与手术拓扑问题，包括一个更高的维Smale猜想。在与Aaron Naber的联合项目中，PI将证明Ricci流的路径空间的许多估计。事实上，一个演化的黎曼流形族满足这些估计当且仅当它由里奇流演化。在此基础上，PI和Naber将定义Ricci流的弱解的概念，并发展这些弱解的理论。

The project aims to study two geometric versions of the heat equation: the evolution of surfaces by their mean curvature, and the evolution of curved spaces by Ricci flow. Mean curvature flow models many physical processes which involve an evolving surface, or interface. It is the most efficient way to decrease the area of surfaces and to evolve them towards optimal ones. Correspondingly, Ricci flow deforms curved spaces towards optimal shapes. While many foundational results have been obtained on both flows, a central problem is that in most relevant situations singularities will form. The proposed research of the PI will provide ways to understand these singularities and to continue the flow through them. This will facilitate many new applications both within and outside mathematics.The proposed research is on mean curvature flow and Ricci flow, with a focus on the formation of singularities and techniques to continue the flow through singularities. In a joint work with Bruce Kleiner, the PI will give a new construction of mean curvature flow with surgery, based on their recent estimates for mean convex flows. The new construction is both more elementary and substantially shorter than prior ones. The PI will apply the mean curvature flow with surgery to topological problems, including a higher dimensional Smale conjecture. In a joint project with Aaron Naber, the PI will prove numerous estimates on path space for the Ricci flow. In fact, an evolving family of Riemannian manifolds satisfies these estimates if and only if it evolves by Ricci flow. Based on this, the PI and Naber will define a notion of weak solutions for the Ricci flow, and develop the theory of these weak solutions.