Mean curvature flow and geometric analysis

平均曲率流和几何分析

• 批准号: 1206827
• 负责人: William Minicozzi
• 金额: $ 72.66万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2014-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206827&HistoricalAwards=false
• 关键词: Mean; curvature; flow; geometric; analysis

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.

PI提议与Toby Colding一起继续他们对平均曲率流和相关几何分析领域的研究，包括爱因斯坦流形切锥的唯一性项目。该提案的第一个广泛领域集中在对MCF中的奇点的研究上，包括奇异集大小的估计，奇点可能类型的紧性定理，流的一般奇点的分类，以及描述在一般奇点的邻域中的流的规范邻域定理。这些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成果。该建议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年提出的，他们在PI想要去除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都扮演着重要的角色，对于理解爱因斯坦流形极限的奇异集的结构也非常重要。这个项目的重点是几个几何变分问题。这些问题都是数学上的，但其中许多问题首先出现在科学和工程领域。也许最自然的几何形状是最小的表面，局部面积最小，因此，模型肥皂膜处于完美的平衡状态（所以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，人们就开始研究这些问题，但近年来，在许多数学家的重要贡献下，极小曲面理论中许多长期存在的问题取得了突破。这有一个时变的模拟，即一个表面（不处于平衡状态）演变成尽可能快地最小化其表面积；这被称为平均曲率流或MCF。从数学上讲，这导致了一个非线性偏微分方程，它在形式上类似于物理学中控制热量流动的方程。显然，最小的表面在MCF下保持静态。MCF和其他几何流因其内在之美以及它们在其他领域的潜在应用而发展起来，例如期权定价、退火金属中的晶粒运动和晶体生长。虽然已经获得了关键的基础结果，但几个最基本的问题仍未得到解答。相反，爱因斯坦方程是空间（或广义相对论中的时空）曲率的非线性微分方程。希尔伯特在一个世纪前就意识到这是爱因斯坦-希尔伯特泛函的变分欧拉-拉格朗日方程。