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Analysis of Singularities of the Ricci Flow

里奇流的奇点分析

基本信息

批准号：
1811845
负责人：
Ovidiu Munteanu
金额：
$ 15.82万
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：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811845&HistoricalAwards=false
关键词：
Analysis Singularities Ricci Flow

项目摘要

Differential geometry is the key mathematics in Einstein's theory of relativity. Indeed, Einstein wrote his famous field equations in the language of tensors on manifolds, and these equations have since been well studied by mathematicians and physicists alike. Certainly, besides being fundamental in general relativity, differential geometry is also very useful in other fields of science, such as in control theory, in computer vision, data analysis, and many others. For this reason, understanding the structure of manifolds is a fundamental problem in science. This project will focus on the behavior of geometric flows on manifolds. A typical example of a geometric flow is the heat equation, which describes the distribution of heat in a region over time. This proposal studies a more advanced form of the heat equation, called the Ricci flow. A Riemannian metric on a manifold tells us about the shape of that object, how to measure angles and distances. The Ricci flow is a heat-type equation for Riemannian metrics. It is hoped, and confirmed in some cases, that the Ricci flow will evolve a given metric on a manifold to an improved one, such as an Einstein metric. However, a major difference between this theory and that of the standard heat equation is that the Ricci flow is a non-linear equation, and as such it usually develops singularities after some time. When such singularities are understood, the process may be continued. This has played a central role in the proof of the long-standing Poincare conjecture about the topology of three dimensional manifolds. The main goal of this project is to understand such singularities in dimension four, and to investigate the implications of our findings to the structure of four dimensional manifolds. Because Ricci flow can be seen as the renormalization group flow in string theory, there are other possible applications of this study to theoretical physics. Other related flows, like the 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 the development of young talent.Ricci flow was introduced by Richard 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 lose 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 scaling. Three-dimensional shrinking Ricci solitons have been classified through the work of Hamilton, Ivey and Perelman. This has important consequences to understanding the behavior of Ricci flow with surgeries on three-dimensional manifolds, and indeed, for the resolution of the Poincare conjecture. The main goal of this project is to classify four-dimensional complete noncompact Ricci solitons. This will be achieved through a complete understanding of the asymptotic geometry of these spaces and through studying corresponding rigidity questions. It is expected that this project will advance our insight on the behavior of Ricci flow in dimension four, which will enable a Ricci flow approach to some important questions about the topology of four dimensional manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

微分几何是爱因斯坦相对论中的核心数学。事实上，爱因斯坦用流形上的张量语言写下了他著名的场方程，自那以后，数学家和物理学家都对这些方程进行了很好的研究。当然，除了广义相对论的基本原理外，微分几何在其他科学领域也非常有用，如控制论、计算机视觉、数据分析等。因此，理解流形的结构是科学中的一个基本问题。这个项目将集中在流形上的几何流的行为。几何流动的一个典型例子是热量方程，它描述了一个区域内热量随时间的分布。这一提议研究了一种更高级的热方程形式，称为Ricci流。流形上的黎曼度量告诉我们物体的形状，如何测量角度和距离。Ricci流是黎曼度规的热型方程。人们希望并在某些情况下证实，Ricci流将把流形上的给定度量演化为改进的度量，例如爱因斯坦度量。然而，这一理论与标准热方程的主要不同之处在于，Ricci流是一个非线性方程，因此它通常在一段时间后出现奇点。当这样的奇点被理解时，这个过程就可以继续下去。这在证明长期存在的关于三维流形拓扑的庞加莱猜想中发挥了核心作用。这个项目的主要目标是理解四维空间中的这种奇点，并研究我们的发现对四维流形结构的影响。因为Ricci流可以看作弦理论中的重整化群流，所以这项研究在理论物理中还有其他可能的应用。其他相关的流动，如平均曲率流，在其他领域有更显著的应用，如在计算机可视化、消除噪声或冶金、金属热处理方面。这个项目的推广部分将结果传播给普通公众，并有助于年轻人才的发展。Ricci Flow是由Richard Hamilton在80年代初引入的，这是一项致力于理解正曲线三维流形的基础工作。后来变得很清楚，如果一个人在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成奇点。人们需要了解这些奇点才能继续流动，并且不会丢失关于空间的任何重要的拓扑信息。Ricci流的奇点用Ricci孤子来描述，这些孤子是流的不动点、模微分同胚和标度。哈密尔顿、艾维和佩雷尔曼对三维收缩的利玛窦孤子进行了分类。这对于理解三维流形上外科手术的Ricci流的行为，甚至对于解决Poincare猜想都有重要的意义。这个项目的主要目标是对四维完全非紧致Ricci孤子进行分类。这将通过完全理解这些空间的渐近几何并通过研究相应的刚性问题来实现。预计这个项目将促进我们对四维空间中Ricci流行为的洞察，这将使Ricci流方法能够解决关于四维流形拓扑的一些重要问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。