Geometric Flows, Geometric Inequalities, and Rigidity of Embeddings

几何流、几何不等式和嵌入刚性

基本信息

批准号：
2103573
负责人：
Simon Brendle
金额：
$ 22.16万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2103573&HistoricalAwards=false
关键词：
Geometric Flows Inequalities Rigidity Embeddings

项目摘要

This project is focused on questions in differential geometry. The aim of differential geometry is to study higher-dimensional shapes and their curvature. In particular, these concepts provide a mathematical framework for general relativity. Geometric flows are a key tool in differential geometry. The idea here is to take a geometric object and evolve it by a differential equation in order to smooth it out. These differential equations share common features with heat diffusion. However, the differential equations that arise in geometry tend to be nonlinear, which presents challenges in their analysis. A main focus is to understand the behavior of these equations when the solution becomes singular, that is, when the curvature becomes very large. An important problem is to classify the singularity models; these are the limiting shapes that occur at a singularity. Another major goal in geometry is to understand geometric inequalities. A basic example is the isoperimetric inequality (which states that balls have smallest surface area among all shapes that enclose a given amount of volume), but many other types of inequalities are of importance in differential geometry. The project also includes training of PhD students and mentoring of post-doctoral researchers.The primary examples of geometric flows are the Ricci flow and the mean curvature flow. The mean curvature flow is the most natural evolution equation for a surface embedded in Euclidean space, while the Ricci flow is the most natural evolution equation for a Riemannian metric. The Ricci flow has become an indispensable tool in differential geometry. Among other things, the Ricci flow lies at the heart of Perelman's proof of the Poincare conjecture. The PI will study what types of singularities can form under these evolution equations. For example, it would be very interesting to understand whether a plane of multiplicity 2 can arise as a singularity model under the mean curvature flow. In another direction, the PI will study problems related to geometric inequalities. In particular, it would be very interesting to understand isoperimetric inequalities in negatively curved manifolds. Moreover, many geometric inequalities come with an associated rigidity statement which characterizes the case of equality. The PI will study the near-equality case in such inequalities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的重点是差异几何形状的问题。差异几何形状的目的是研究更高维的形状及其曲率。特别是，这些概念为一般相对性提供了数学框架。几何流程是差异几何形状的关键工具。这里的想法是采用几何对象并通过微分方程将其进化，以使其平滑。这些微分方程在热扩散中具有共同的特征。但是，几何形状中出现的微分方程往往是非线性的，它在其分析中提出了挑战。一个主要重点是了解当解决方案变得单数时，即曲率变得非常大时，这些方程式的行为。一个重要的问题是对奇异模型进行分类。这些是以奇异性出现的极限形状。几何学的另一个主要目标是了解几何不平等现象。一个基本的例子是等等的不平等（指出，所有形状之间的球具有最小的表面积，它们封闭了给定数量的体积），但是许多其他类型的不平等现象在差异几何形状中很重要。该项目还包括对博士生的培训和指导博士后研究人员。几何流量的主要例子是RICCI流量和平均曲率流量。平均曲率流是嵌入欧几里得空间中表面的最自然进化方程，而RICCI流量是Riemannian公制的最自然进化方程。 RICCI流已成为差异几何形状中必不可少的工具。除其他事项外，RICCI的流动是Perelman庞贝拉猜想证明的核心。 PI将研究这些演化方程式可以形成哪些类型的奇异性。例如，了解多样性2的平面是否可以作为平均曲率流下的奇异模型出现，这将非常有趣。在另一个方向上，PI将研究与几何不平等相关的问题。特别是，在负弯曲的流形中了解等等的不平等现象将非常有趣。此外，许多几何不等式都带有相关的刚度声明，其特征是平等的情况。 PI将在这种不平等现象中研究近平等的案例。该奖项反映了NSF的法定任务，并认为使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估。