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.

里奇流是一个几何过程，可以用来平滑，有时均匀，一个给定的空间。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，例如庞加莱猜想和三维空间中的几何化猜想。在利玛窦流的研究中，一般的期望是流在极限中产生一种几何，这种几何在某种程度上是拓扑所固有的，即底层空间的松散构成。然而，大多数情况下，流动会产生某些奇点，在流动可以继续之前，必须通过所谓的“手术”来去除这些奇点。尽管它们具有强大的拓扑意义，但在三维或更高维中，带手术的里奇流仍然没有得到很好的理解。这个项目的目标是获得一个更好的理解的长期行为的三维Ricci流手术，和依赖于初始条件的演变的几何形状。此外，还提出了研究高维Ricci流的建议。第一个项目是关于外科手术的三维Ricci流的长时间行为的分析。该项目建立在主要研究者以前的工作基础上，其中建立了手术数量的有限性，并导出了流量的长时间渐近性的初始描述。第二个项目的目标是从一个给定的黎曼度量连续族出发，构造带手术的Ricci流连续族。在这样的家庭中，手术可以根据参数在空间和时间上连续移动，它们可能出现或消失。一个成功的建设，这样的家庭可以最有可能被用来解决一个猜想，指出空间的正标量曲率度量的3球是可收缩的。此外，它还可以用来解决广义Smale猜想，该猜想对球面3-流形的同构群的拓扑进行了分类。在第三个项目中，主要研究者提出了与有界标量曲率的Ricci流研究相关的几个问题的工作。这项研究是与张琦合作进行的前期工作的延续。建议的问题包括有界标量曲率的四维Ricci流的奇异性分析，和非崩溃的，长期存在的Ricci流，特别是在4维的研究。