Geometric flows and four-dimensional geometry

几何流和四维几何

• 批准号: 1301864
• 负责人: Jeffrey Streets
• 金额: $ 14.59万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301864&HistoricalAwards=false
• 关键词: Geometric; flows; four; dimensional; geometry

This project lies at the interface of geometry and partial differential equations. The PI will further the study of natural geometric evolution equations designed to understand aspects of four dimensional geometry and topology. One project is to understand the singularity formation and long time behavior of geometric flows on almost Hermitian manifolds, previously introduced by the PI in joint work with G. Tian. One aspect of this is to fully understand how the long time behavior of the pluriclosed flow relates to the classification of complex surfaces. Another project is to understand the subcritical behavior of the gradient flow of the Yang-Mills energy of a Riemannian metric. One major goal here is to understand the possible singularities which can develop on four-manifolds with small initial energy.The use of geometry and curvature in understanding spaces is a fundamental idea in mathematics and science. Frequently the relevant curvature properties are closely related to equations relating to the laws of nature. Geometric evolution equations of the kind proposed here arise frequently in various physical contexts, and see application to various areas including image analysis and material science. The proposed research will advance fundamental geometric and analytic ideas which are the foundation of such applications.

本课题是几何与偏微分方程的结合。PI将进一步研究自然几何演化方程，旨在了解四维几何和拓扑结构的各个方面。其中一个项目是了解几乎厄米流形上几何流动的奇点形成和长时间行为，之前由PI在与G. Tian的联合工作中引入。其中一个方面是充分理解多闭流的长时间行为与复杂表面分类的关系。另一个项目是了解黎曼度规的杨-米尔斯能量梯度流动的亚临界行为。这里的一个主要目标是理解在初始能量小的四流形上可能出现的奇点。利用几何和曲率来理解空间是数学和科学的一个基本思想。通常，相关的曲率性质与有关自然规律的方程密切相关。这里提出的几何演化方程经常出现在各种物理环境中，并应用于包括图像分析和材料科学在内的各个领域。提出的研究将推进基本的几何和分析思想，这是这些应用的基础。