Mean curvature flow and geometric analysis

PI建议与Toby Colding一起继续他们对平均曲率流和几何分析相关领域的研究，包括关于爱因斯坦流形的切锥唯一性的项目。该建议的第一个广泛领域集中于对MCF中奇点的研究，包括对奇点集大小的估计、关于可能的奇点类型的紧致性定理、流的类属奇点的分类以及描述类属奇点邻域中的流的规范邻域定理。这些都是关于奇点的最重要的问题，PI和他的合作者已经在这个方向上取得了重要的结果。该提案的第二个主要领域涉及爱因斯坦流形的结构，特别是爱因斯坦流形何时具有唯一的渐近结构(或无穷远处的切锥)的问题。在这个方向上的主要先验结果是由Cheeger和Tian在1994年得到的，其中他们在PI想要去掉的可积性假设下证明了唯一性。这类唯一性问题在几何分析的许多领域中起着重要的作用，并且对于理解爱因斯坦流形极限的奇异集的结构是极其重要的。这些问题是数学问题，但其中许多问题首先出现在科学和工程领域。也许在几何上最自然的是局部最小化其表面积的极小表面，从而使肥皂膜处于完美平衡状态(因此，膜在其他时间不会改变)。至少从拉格朗日1762年的回忆录开始，人们就已经对这些问题进行了研究，但近年来，极小曲面理论中的许多长期存在的问题都取得了突破，许多数学家做出了重要贡献。有一个与此类似的时变模拟，其中一个曲面(不处于平衡状态)进化以尽可能快地最小化其表面积；这称为平均曲率流或MCF。在数学上，这导致了一个形式上类似于物理中控制热流的方程的非线性偏微分方程式。显然，最小曲面在MCF下保持静态。MCF和其他几何流是因为它们的内在美以及它们在其他领域的潜在应用而被开发出来的，例如，期权定价、退火金属中的颗粒运动和晶体生长。虽然已经取得了关键的基础性成果，但仍有几个最基本的问题没有得到回答。相比之下，爱因斯坦方程是一个关于空间(或广义相对论中的时空)曲率的非线性微分方程。一个世纪前，希尔伯特意识到，这就是爱因斯坦-希尔伯特泛函的欧拉-拉格朗日方程。

The PI proposes, jointly with Toby Colding, to continue their investigations on mean curvature flow and related areas of geometric analysis, including projects on the uniqueness of tangent cones for Einstein manifolds. The first broad area of the proposal centers on the study of singularities in MCF, including estimates for the size of the singular sets, a compactness theorem for the possible types of singularities, a classification of generic singularities of the flow, and a canonical neighborhoods theorem that describes the flow in a neighborhood of the generic singularities. These are the most important questions about singularities and the PI and his collaborator have already obtained significant results in this direction. The second main area of the proposal concerns the structure of Einstein manifolds and, in particular, the question of when an Einstein manifold has a unique asymptotic structure (or tangent cone at infinity). The main prior result in this direction is due to Cheeger and Tian in 1994, where they showed uniqueness under an integrability assumption that the PI would like to remove. These sort of uniqueness questions have played an important role in a number of areas of geometric analysis and are extremely important in understanding the structure of the singular set for limits of Einstein manifolds.This project focuses on several geometric variational problems. The problems are mathematical, but many of them arose first in science and engineering. Perhaps the most natural geometrically are minimal surfaces that locally minimize their surface area and, thus, model soap films in perfect equilibrium (so the film does not change other time). These have been studied at least since Lagrange's 1762 memoir, but recent years have seen breakthroughs on many long-standing problems in the theory of minimal surfaces, with important contributions from many mathematicians. There is a time-varying analog of this where a surface (which is not in equilibrium) evolves to minimize its surface area as quickly as possible; this is called mean curvature flow or MCF. Mathematically, this leads to a nonlinear partial differential equation which is formally similar to the equation that governs the flow of heat in physics. Clearly, minimal surfaces remain static under the MCF. MCF and other geometric flows were developed for their intrinsic beauty as well as their potential applications to other fields to model, for instance, option pricing, motion of grains in annealing metals, and crystal growth. While key foundational results have been obtained, several of the most basic questions remain unanswered. In contrast, the Einstein equation is a nonlinear differential equation for the curvature of a space (or a space-time in general relativity). Hilbert realized a century ago that this comes up variationally as the Euler-Lagrange equation for the Einstein-Hilbert functional.