权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Geometric flows on Riemannian and Kaehler manifolds

黎曼流形和凯勒流形上的几何流

基本信息

批准号：
1105549
负责人：
Lei Ni
金额：
$ 14.85万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2011
资助国家：
美国
起止时间：
2011-08-15 至 2014-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105549&HistoricalAwards=false
关键词：
Geometric flows Riemannian Kaehler manifolds

项目摘要

The first part of the project is about the geometric and analytic properties of the Ricci flow equation and their applications to the study of geometry. Namely to study the classifications of self-similar solutions, e.g gradient solitons and ancient solutions, as well as the convexity type estimates in general. Such results have far-reaching consequences in the singularity analysis, and applications of Ricci flow in the study of geometric-topological structure of the manifolds. The second theme of the project is on the sharp gradient estimates of Li-Yau-Hamilton type, related monotonicity formulae and applications in geometric nonlinear PDEs. Their relations to physics, statistical mechanics will be studied too. The aim is to discover a fundamental physical/geometric principle to unify various sharp estimates and monotonicity formulae. It will also provide the guideline for further discovery of the new monotonicity formulae in other geometric PDEs.Since all physical event takes place in a space, the subject of differential geometry which studies the geometric properties of the space has important consequence in every physical event. This project mainly involves the study of partial differential equations of parabolic type which arise from differential geometry, and their applications to the understanding of various geometric/topological properties of manifolds. This area lies in the center of the current mathematics. It naturally connects various area of mathematics, such as topology, Riemannian geometry, partial differential geometry, Lie groups, as well as mathematical physics. The techniques developed can be useful in understanding problems in economics, material sciences and bio-sciences.

第一部分是关于Ricci流方程的几何和解析性质及其在几何研究中的应用。即研究自相似解的分类，如梯度孤子和古解，以及一般的凸型估计。这些结果在奇点分析以及Ricci流在流形的几何拓扑结构研究中的应用方面具有深远的意义。第二个主题是Li-Yau-汉密尔顿型的锐梯度估计，相关的单调性公式及其在几何非线性偏微分方程中的应用。同时也将研究它们与物理学、统计力学的关系。目的是发现一个基本的物理/几何原理，统一各种尖锐的估计和单调性公式。这也将为进一步发现其他几何偏微分方程的新单调性公式提供指导。由于所有的物理事件都发生在一个空间中，研究空间几何性质的微分几何学科在任何物理事件中都有重要的意义。本专题主要研究微分几何中的抛物型偏微分方程及其在理解流形的各种几何/拓扑性质中的应用。这一领域是当前数学的中心。它自然地连接了数学的各个领域，如拓扑学、黎曼几何、偏微分几何、李群以及数学物理。所开发的技术可用于理解经济学、材料科学和生物科学中的问题。