On the long-time behavior of Ricci flow and Ricci flow surgery

论Ricci流和Ricci流手术的长期行为

• 批准号: 1611906
• 负责人: Richard Bamler
• 金额: $ 17.4万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1611906&HistoricalAwards=false
• 关键词: long; time; behavior; Ricci; flow

A Ricci flow is a geometric process that can be used to smooth out, and sometimes homogenize, a given space. Its mathematical significance has become apparent by the fact that it could be used to prove various conjectures, such as the Poincaré and Geometrization Conjectures in 3-dimensional spaces. A general expectation in the study of Ricci flows is that the flow produces a geometry in the limit that is somehow inherent to the topology, i.e. the loose makeup, of the underlying space. Most often, however, the flow develops certain singularities, which have to be removed by so-called "surgeries" before the flow can be continued. Despite their powerful topological implications, Ricci flows with surgery are still not well understood in dimensions 3 or higher. The goal of this project is to obtain a better understanding of the long-time behavior of 3 dimensional Ricci flows with surgery, and the dependence of the evolved geometries on initial conditions. Moreover, the study of Ricci flows in higher dimensions is suggested.The proposal is split into three projects. The first project concerns the analysis of the long-time behavior of 3 dimensional Ricci flows with surgery. This project builds on previous work of the principal investigator, in which the finiteness of the number of surgeries was established and in which an initial description of the flow's long-time asymptotics was derived. The objective of the second project is to construct continuous families of Ricci flows with surgery, starting from a given continuous family of Riemannian metrics. In such families, surgeries may move continuously in space and time depending on the parameter, and they may appear or disappear. A successful construction of such families can most likely be used to solve a conjecture that states that the space of positive scalar curvature metrics on the 3-sphere is contractible. Moreover, it may be used to solve the Generalized Smale Conjecture, which classifies the topology of diffeomorphism groups of spherical 3-manifolds. In the third project, the principal investigator proposes the work on several problems associated with the study of Ricci flows with bounded scalar curvature. This study is a continuation of previous work conducted in collaboration with Qi Zhang. The suggested problems include the analysis of singularities in 4-dimensional Ricci flows with bounded scalar curvature, and the study of non-collapsed, long-time existent Ricci flows, especially in dimension 4.

里奇流是一种几何过程，可以用来平滑，有时甚至均匀化给定的空间，它的数学意义已经变得显而易见，因为它可以用来证明各种猜想，例如庞加莱猜想和几何化猜想。 3 维空间。利玛窦流研究中的一个普遍期望是，流在某种程度上产生了拓扑所固有的几何形状，即底层空间的松散结构。然而，流动常常会产生某些奇点，在流动继续之前必须通过所谓的“手术”去除这些奇点，尽管它们具有强大的拓扑意义，但在 3 维或更高维度中，手术中的利玛窦流仍然没有得到很好的理解。该项目的目标是更好地了解手术中 3 维 Ricci 流的长期行为，以及演化几何形状对初始条件的依赖性。此外，建议研究更高维度的 Ricci 流。提议分为三个项目。第一个项目涉及手术中 3 维 Ricci 流的长期行为的分析。该项目建立在主要研究者之前的工作基础上，其中确定了手术数量的有限性。第二个项目的目标是从给定的黎曼度量的连续族开始，通过手术构建连续的 Ricci 流族。在这些族中，手术可以连续移动。空间和时间取决于参数，并且它们可能会出现或消失。此类族的成功构建很可能用于解决 3 球面上的正标量曲率度量的空间是可收缩的猜想。它可用于解决广义 Smale 猜想，该猜想对球面 3-流形的微分同胚群的拓扑进行分类。在第三个项目中，主要研究者提出了与该猜想相关的几个问题的工作。具有有界标量曲率的 Ricci 流的研究 这项研究是与 Qi Zhang 合作进行的先前工作的延续。建议的问题包括具有有界标量曲率的 4 维 Ricci 流的奇点分析以及非塌缩的研究。 ，长期存在的 Ricci 流，特别是在维度 4 中。