The geometry of Ricci solitons

里奇孤子的几何结构

• 批准号: 1506220
• 负责人: Ovidiu Munteanu
• 金额: $ 16.65万
• 依托单位: University of Connecticut
• 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: 2015-07-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1506220&HistoricalAwards=false
• 关键词: geometry; Ricci; solitons

Differential geometry is a branch of mathematics that studies the shapes of geometric objects, called manifolds. Differential geometry is the key mathematics in Einstein's theory of relativity, being used to describe the curvature of space-time in the presence of a body of mass and energy. As such, understanding the geometry and topology of manifolds is a fundamental problem in science. This project will focus on studying the behavior of evolution equations on manifolds. Geometric flows have proved to be very important in mathematics, in particular, Ricci flow has been very popular for its use in the resolution of some central problems, such as the long standing Poincare conjecture. Geometric flows can be used to study a fundamental question of geometry, which is to find canonical metrics on a given manifold. The Ricci flow proposes to do this in an analytic way, by flowing a given metric in time towards an improved, canonical one. This proposal will investigate the formation of singularities along the flow and will attempt to understand them in dimension four. There are possible applications of this study to theoretical physics, because Ricci flow can be seen as the renormalization group flow in string theory. Other related flows, like mean curvature flow, have further remarkable applications to other fields, such as in computer visualization, for eliminating noise, or in metallurgy, for heat treatment of metals. The outreach components of this project disseminate the results to general public and contribute to developing of young talent.Ricci flow was introduced by Hamilton in the early eighties, in a fundamental work devoted to understanding positively curved three dimensional manifolds. It became clear later that if one flows an arbitrary metric on a given manifold, the flow will generally develop singularities. One needs to understand these singularities in order to continue the flow, and to not loose any significant topological information about the space. The singularities of Ricci flow are modeled by Ricci solitons, which are fixed points of the flow, modulo diffeomorphisms and scalings. Perelman classified three dimensional shrinking Ricci solitons, and used this classification in the resolution of the Poincare conjecture. This has attracted much attention on the higher dimensional problem, and its applications. The goal of this project is to understand the structure and properties of Ricci solitons, in arbitrary dimension. This will lead to a better understanding of how large the space of Ricci solitons is. The study will focus in particular on complete four dimensional shrinking Ricci solitons. There are significant new challenges to this problem, in particular, there exist higher dimensional examples of Ricci solitons which are not positively curved. This principal investigator will attempt to understand the asymptotic geometry of complete four dimensional shrinking Ricci solitons, which is key information for their ultimate classification.

微分几何是数学的一个分支，它研究被称为流形的几何物体的形状。微分几何是爱因斯坦相对论中的关键数学，用于描述存在质量和能量的物体时的时空曲率。因此，理解流形的几何和拓扑是科学中的一个基本问题。本计画将著重于研究流形上演化方程的行为。几何流在数学中有着重要的地位，特别是Ricci流，它在解决一些中心问题上有着广泛的应用，如庞加莱猜想。几何流可以用来研究几何学的一个基本问题，即在给定流形上寻找正则度量。利玛窦流建议以分析的方式做到这一点，通过将给定的度量及时流向改进的规范度量。这个建议将调查形成的奇点沿着流动，并将试图了解他们在第四维。这项研究可能应用于理论物理，因为里奇流可以被看作是弦理论中的重整化群流。其他相关的流动，如平均曲率流动，在其他领域也有显著的应用，如在计算机可视化中，用于消除噪音，或在冶金学中，用于金属的热处理。该项目的推广部分向公众传播成果，并有助于培养青年人才。里奇流介绍了汉密尔顿在八十年代初，在一个基本的工作，致力于了解积极弯曲的三维流形。后来变得清楚的是，如果一个人在一个给定的流形上流动一个任意的度量，那么这个流动通常会发展出奇点。人们需要理解这些奇点，以便继续流动，并且不丢失关于空间的任何重要拓扑信息。Ricci流的奇异性由Ricci孤子来描述，它们是流的不动点、模同态和标度。Perelman对三维收缩Ricci孤子进行了分类，并将这种分类用于解决庞加莱猜想。这引起了人们对高维问题及其应用的广泛关注。本项目的目标是了解任意维的Ricci孤子的结构和性质。这将有助于我们更好地理解Ricci孤子的空间有多大。研究将特别集中在完整的四维收缩Ricci孤子。这一问题面临着新的挑战，特别是存在着非正弯曲的高维Ricci孤子的例子。本文主要研究完全四维收缩Ricci孤子的渐近几何，这是它们最终分类的关键信息。