Variational and Parabolic Phenomena in Differential Geometry

微分几何通过距离和角度研究空间的形状。从数学上讲，曲率的概念起着核心作用。偏微分方程组是由微分几何中的许多基本问题自然产生的，特别是在理解几何空间的曲率或寻找给定空间上的最优几何结构时。PI将集中于两个重要的例子：极小子流形和几何流。极小子流形是局部最小化面积(或体积)的子空间。自从欧拉和拉格朗日的工作以来，人们一直在研究它们，并在几何学和其他相关领域产生了许多应用，如低维拓扑和广义相对论。此外，它们是自然界中许多有趣的非线性现象的重要模型，在他们的研究中形成的各种想法被证明是变分、几何偏微分方程组和数学物理中的关键。另一方面，几何流是通过来自曲率的抛物型偏微分方程组来变形给定几何图形的过程。一个突出的例子是由Ricci曲率产生的流，或称Ricci流，它对汉密尔顿和佩雷尔曼的工作对三维空间的几何化产生了开创性的影响。这个研究项目的一个目的是，某些拓扑型和Morse指数型极小子流形的存在可以对环境流形的曲率施加限制，反之亦然。PI将研究存在正曲率时极小子流形的Morse指数和拓扑的几个问题。另一个方面是几何流中的奇点形成现象，重点研究了奇点的轮廓及其刚性和稳定性。

Differential Geometry studies shapes of spaces through distances and angles. Mathematically the concept of curvature plays the central role. Partial differential equations (PDEs) arise naturally from many fundamental questions in Differential Geometry, notably in understanding a geometric space in terms of its curvature or finding an optimal geometric structure on a given space. The PI will focus on two important examples: minimal submanifolds and geometric flows. Minimal submanifolds are subspaces that locally minimize area (or volume). They have been studied since the work of Euler and Lagrange, yielding many applications to geometry and other related fields such as low-dimensional topology and general relativity. Moreover, they are important models for many interesting non-linear phenomena in nature and a variety of ideas developed in their study have turned out to be key in the calculus of variations, geometric PDEs, and mathematical physics. A geometric flow, on the other hand, is a process of deforming a given geometry through a parabolic system of PDEs coming from curvature. A prominent example is the flow by the Ricci curvature, or Ricci flow, which has had seminal consequences to the geometrization of three-dimensional spaces by the work of Hamilton and Perelman. One of the tenets of this research project is that the existence of minimal submanifolds of certain topological and Morse index type can impose restrictions on the curvature of the ambient manifold, and vice-versa. The PI will investigate several questions relating the Morse index and the topology of minimal submanifolds in presence of positive curvature. Another aspect is the singularity formation phenomena in geometric flows, with a focus on the profiles of singularities and their rigidity and stability properties.