Mean curvature flow and Ricci flow

该项目旨在研究热方程的两个几何版本：通过曲面的平均曲率来演化曲面，以及通过Ricci流来演化曲面空间。平均曲率流模拟了许多涉及不断变化的表面或界面的物理过程。这是减少曲面面积并使其向最佳曲面演化的最有效方法。相应地，Ricci流将曲线空间变形为最佳形状。虽然在这两种流动上都取得了许多基础性的结果，但一个中心问题是，在大多数相关的情况下，会形成奇点。对PI的拟议研究将提供理解这些奇点并继续流经它们的方法。这将促进数学内外的许多新的应用。拟议的研究是关于平均曲率流和Ricci流，重点是奇点的形成和通过奇点继续流动的技巧。在与Bruce Kleiner的合作中，PI将根据他们最近对平均凸流的估计，通过手术给出平均曲率流的新结构。新的结构比以前的结构更基本，也大大缩短了。PI将通过手术将平均曲率流应用于拓扑问题，包括更高维的斯梅尔猜想。在与Aaron Naber的一个联合项目中，PI将证明Ricci流的路径空间的许多估计。事实上，当且仅当一族黎曼流形通过Ricci流发展时，它才满足这些估计。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.