Variational and Parabolic Phenomena in Differential Geometry

微分几何中的变分和抛物线现象

• 批准号: 1512574
• 负责人: Davi Maximo
• 金额: $ 14.64万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2017-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1512574&HistoricalAwards=false
• 关键词: Variational; Parabolic; Phenomena; Differential; Geometry

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.

微分几何通过距离和角度来研究空间的形状。在数学上，曲率的概念起着核心作用。偏微分方程（PDE）自然地产生于微分几何中的许多基本问题，特别是在理解几何空间的曲率或在给定空间上找到最佳几何结构。PI将专注于两个重要的例子：极小子流形和几何流。极小子流形是局部极小化面积（或体积）的子空间。自从欧拉和拉格朗日的工作以来，它们一直在研究，在几何和其他相关领域，如低维拓扑学和广义相对论中产生了许多应用。此外，它们是自然界中许多有趣的非线性现象的重要模型，在它们的研究中发展的各种想法已经成为变分法，几何偏微分方程和数学物理的关键。另一方面，几何流是通过来自曲率的偏微分方程的抛物系统使给定几何形状变形的过程。一个突出的例子是里奇曲率流，或里奇流，它对汉密尔顿和佩雷尔曼的三维空间的几何化产生了开创性的影响。这个研究项目的原则之一是，存在的极小子流形的某些拓扑和莫尔斯指数类型可以施加限制的曲率的环境流形，反之亦然。PI将研究几个问题有关的莫尔斯指数和拓扑的极小子流形在存在的正曲率。另一方面是几何流中的奇点形成现象，重点是奇点的轮廓及其刚度和稳定性。