Ricci Flows through Singularities and Ricci Flows with Bounded Scalar Curvature

穿过奇点的里奇流和具有有界标量曲率的里奇流

基本信息

批准号：
1906500
负责人：
Richard Bamler
金额：
$ 44.15万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906500&HistoricalAwards=false
关键词：
Ricci Flows through Singularities Bounded

项目摘要

A Ricci flow is a geometric process that may be used to improve a given geometry towards a more homogeneous one. Ricci flows have become the subject of intensive study, as they have been used to prove various long-standing conjectures, such as the Poincare and Geometrization Conjectures in dimension 3. The general expectation is that a Ricci flow produces a geometry in the limit that is in some sense inherent to the topology, i.e. the loose makeup, of the underlying space. However, usually a Ricci flow develops complicated singularities in finite time. In dimension 3 these singularities can be removed manually by so called surgeries and the flow can be continued beyond them. Recently, a new class of "singular Ricci flows" was introduced in dimension 3. These flows flow "automatically through singularities at an infinitesimal scale", thereby eliminating the somewhat unnatural surgery process. The goal of this project is to understand these flows further and to use this understanding to study the topology of certain spaces of metrics and diffeomorphism groups. In addition, the PI will work on Ricci flows in higher dimensions, aimed at understanding their singularity formation, which may result in a similar surgery or singular flow construction.The research project is split into two projects. The first project is a continuation of the PI's work (in collaboration with Bruce Kleiner) on the uniqueness and continuity of singular Ricci flows through singularities. The general goal of this project is to understand the geometric, topological and analytic applications of this work. Among other things, the PI has a strategy to resolve the Generalized Smale Conjecture, which would extend a previous partial resolution by the PI and Kleiner. Further potential applications concern the topology and geometry of the space of positive scalar curvature metrics, as well as the study of generic Ricci flows in dimension 3. The second project is a continuation of the PI's work on the study of Ricci flows with bounded scalar curvature. The PI will investigate several conjectures that have been verified under the assumption of bounded scalar curvature. These conjectures are likely to remain true if this assumption is removed.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.

RICCI流是一种几何过程，可用于改善给定的几何形状更均匀的几何形状。 RICCI流量已成为强化研究的主题，因为它们已被用来证明各种长期的猜想，例如维度3中的Poincare和几何化猜想。普遍的期望是RICCI流动在极限上产生几何形状，而在某种程度上是拓扑所固有的，即较宽的构成，即底层较宽的构成。但是，通常在有限的时间内，RICCI流程会形成复杂的奇异点。在维度3中，可以通过所谓的手术手动删除这些奇点，并且可以继续进行流动。最近，在Dimension 3中引入了一类新的“单一RICCI流”。这些流以无限尺度自动通过奇点“自动流动”，从而消除了某种不自然的手术过程。该项目的目的是进一步了解这些流量，并利用这种理解来研究指标和差异群体的某些空间的拓扑。此外，PI将在更高维度的RICCI流动上工作，旨在了解其奇异性形成，这可能会导致类似的手术或奇异的流动构建。研究项目分为两个项目。第一个项目是Pi的工作（与Bruce Kleiner合作）的延续，涉及奇异Ricci通过奇异性的独特性和连续性。该项目的一般目标是了解这项工作的几何，拓扑和分析应用。除其他事项外，PI具有解决广义的Smale猜想的策略，该猜想将扩大PI和Kleiner先前的部分解决方案。进一步的潜在应用涉及积极标态曲率指标空间的拓扑和几何形状，以及对尺寸3中通用RICCI流的研究。第二个项目是PI在RICCI研究中的延续的RICCI流动研究，具有有限的标量曲率。 PI将研究在有界标态曲率假设下已验证的几种猜想。如果删除此假设，这些猜想可能会保持真实。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来支持。