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.

差异几何形状是研究几何对象的形状，称为歧管的数学分支。差异几何形状是爱因斯坦相对论中的关键数学，用于描述在存在质量和能量体内时时空的曲率。因此，了解流形的几何形状和拓扑是科学中的一个基本问题。该项目将着重于研究流形方程的行为。事实证明，几何流程在数学中非常重要，尤其是Ricci流以解决某些核心问题（例如长期存在的Poincare猜想）中非常流行。几何流程可用于研究几何学的基本问题，即在给定的歧管上找到规范指标。 RICCI的流程提议以分析方式进行此操作，以将给定的度量流向改进的规范。该提议将调查沿流程的奇异性形成，并试图在第四维度中理解它们。这项研究可能应用于理论物理学，因为RICCI流可以看作是字符串理论中的重新归一化组流。其他相关流（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，以用于金属的热处理。该项目的宣传组成部分将结果传播给了公众，并为发展年轻人才的发展做出了贡献。汉密尔顿在八十年代初期引入了RICCI流动，这是一项致力于理解积极弯曲的三维流形的基本工作。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，而不要放弃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 solitons是流量的固定点，Modulo差异性和尺度。佩雷尔曼（Perelman）将三维缩小的ricci孤子子归类，并将这种分类用于解决庞加罗猜想的解决方案。这引起了人们对更高维度问题及其应用的广泛关注。该项目的目的是在任意维度上了解Ricci Solitons的结构和特性。这将使人们更好地了解Ricci Solitons的空间。该研究将特别关注完全四维缩小的Ricci孤子。尤其是这个问题存在重大挑战，尤其是有较高的Ricci孤子示例，这些示例不是积极弯曲的。这位主要研究者将尝试了解完全四维缩小的Ricci Soliton的渐近几何形状，这是其最终分类的关键信息。