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.

差异几何形状是爱因斯坦相对论理论中的关键数学。的确，爱因斯坦用张量的语言写了他著名的野外方程式，此后，这些方程式被数学家和物理学家都很好地研究了。当然，除了是一般相对论的基本性外，差异几何形状在其他科学领域（例如在控制理论，计算机视觉，数据分析等）中也非常有用。因此，了解流形的结构是科学中的一个基本问题。该项目将重点介绍流动上的几何流动的行为。几何流量的一个典型例子是热方程，它描述了随着时间的流逝，热量的分布。该建议研究了一种更先进的热方程式，称为RICCI流。歧管上的Riemannian指标告诉我们有关该物体的形状，如何测量角度和距离的形状。 RICCI流是Riemannian指标的热型方程。希望并确认在某些情况下，RICCI流将在改进的歧管上演变为歧管上的指标，例如爱因斯坦度量标准。但是，该理论与标准热方程的主要区别在于，RICCI流量是非线性方程，因此通常在一段时间后会发展出奇异性。当理解这种奇异性时，可能会继续。这在证明长期以来关于三维流形拓扑的庞加利猜想的证明中发挥了核心作用。该项目的主要目标是了解第四维度中的这种奇异性，并研究我们发现对四个维歧管的结构的含义。因为RICCI流可以看作是弦理论中的重新归一化群体流动，所以这项研究还有其他可能的应用在理论物理学中。其他相关流（例如平均曲率流量）在其他领域（例如在计算机可视化中）具有进一步的应用，以消除噪声或在冶金中用于金属的热处理。该项目的宣传组成部分将结果传播给了公众，并为年轻人才的发展做出了贡献。RicciFlow是由Richard Hamilton在八十年代初引入的，这是一项基本的工作，致力于理解正面弯曲的三维流形。后来很清楚，如果一个人在给定的歧管上流动任意度量，则流动通常会发展出奇异性。一个人需要了解这些奇点才能继续流动，并且不要失去有关该空间的任何重要拓扑信息。 RICCI流的奇异性是由Ricci Soliton建模的，Ricci soliton是流量的固定点，Modulo差异和缩放。汉密尔顿，艾维和佩雷尔曼的工作已经分类了三维缩小的Ricci孤子。这对于理解三维流形的手术的RICCI流动的行为具有重要的后果，而实际上，对于解决了庞加罗的猜想的解决方案。该项目的主要目的是对四维完整的非竞争性RICCI Soliton进行分类。这将通过完全理解这些空间的渐近几何形状以及研究相应的刚性问题来实现。预计该项目将提高我们对RICCI流动在第四维度的行为的见解，这将使RICCI流动方法能够解决有关四维流形拓扑的一些重要问题。该奖项反映了NSF的法定任务，并认为通过基金会的知识分子和更广泛的影响，可以通过评估来进行评估。