Geometric Flows, Geometric Inequalities, and Rigidity of Embeddings

这个项目关注的是微分几何中的问题。微分几何的目的是研究高维形状及其曲率。特别是，这些概念为广义相对论提供了一个数学框架。几何流是微分几何中的一个重要工具。这里的想法是取一个几何对象，并用微分方程对其进行演化，以使其平滑。这些微分方程式具有热扩散的共同特征。然而，几何中出现的微分方程往往是非线性的，这给它们的分析带来了挑战。主要的焦点是了解当解变得奇异时，即当曲率变得非常大时，这些方程的行为。一个重要的问题是对奇点模型进行分类；这些是出现在奇点上的极限形状。几何学的另一个主要目标是理解几何不等式。一个基本的例子是等周不等式(它规定，在封闭给定体积的所有形状中，球的表面积最小)，但许多其他类型的不等式在微分几何中也很重要。该项目还包括对博士生的培训和博士后研究人员的指导。几何流的主要例子是Ricci流和平均曲率流。平均曲率流是欧氏空间中曲面最自然的演化方程，而Ricci流是黎曼度规最自然的演化方程。Ricci流已成为微分几何中不可缺少的工具。其中，利玛奇流是佩雷尔曼证明庞加莱猜想的核心。PI将研究在这些演化方程下可以形成什么类型的奇点。例如，理解在平均曲率流下，重数为2的平面是否可以作为奇点模型出现，这将是非常有趣的。在另一个方向，PI将研究与几何不平等有关的问题。特别是，了解负曲流形中的等周不等式将是非常有趣的。此外，许多几何不等式都伴随着一个刻画等式的刚性陈述。PI将研究这种不平等的近乎平等的情况。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.