Ricci Flow

利玛窦流是一种几何过程，可用于将给定的几何形状改进为更均匀的几何形状。里奇流已经获得了越来越多的兴趣，因为它们已经被用来证明各种长期存在的猜想，如庞加莱猜想和几何化猜想，以及3维的广义小猜想。一般的期望是，里奇流在极限下产生的几何形状在某种意义上是拓扑固有的，也就是说，底层空间的松散组合。然而，里奇流通常在有限时间内产生复杂的奇点。在维度3中，这些奇点可以通过所谓的手术构造手工移除，并且流可以继续超越它们。该项目的长期目标是将这种手术结构推广到4维，甚至更高。为了实现这一目标，PI将使用他最近发现的一种新理论，研究高维里奇流的奇点形成。在第4维上的成功构造可能具有有趣的拓扑和几何应用。PI还将进一步研究利玛奇流与外科手术，以及密切相关的“利玛奇流通过奇点”，在三维空间，并找到进一步的几何和拓扑应用。该奖项为研究生从事与项目相关的研究提供资金。这个研究项目分为两部分。第一个项目是PI最近在高维Ricci流中获得的紧性和部分正则性理论的延续。该项目的目标是利用这一新理论在第4维度构建一个“带手术的里奇流”或“奇点的里奇流”，推广第3维度的类似结构。实现这一目标的策略是推导出爆炸极限的空间渐近估计，并利用这些估计获得奇点形成的定性图像。基于这幅图，下一步是通过圆柱形和圆锥形手术结构去除奇点。该项目还旨在描述流量的长期渐近性。一个成功的构造和分析可能会导致一些有趣的拓扑和几何应用。第二个项目继续PI和合作者的工作，在奇异点的三维奇异里奇流的唯一性和连续依赖性。先前的工作利用这种连续依赖解决了广义小猜想（将某些3-流形的微分同构群划分到同伦）和关于3-流形上具有正标量曲率的度量空间的猜想。该项目将更深入地研究这些证明中使用的技术，这些技术有可能产生进一步的结果。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

A Ricci flow is a geometric process that can be used to improve a given geometry towards a more homogeneous one. Ricci flows have gained increasing interest, as they have been used to prove various longstanding conjectures, such as the Poincaré and Geometrization Conjectures, as well as the Generalized Smale Conjecture 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, that is, the loose makeup, of the underlying space. However, usually a Ricci flow incurs complicated singularities in finite time. In dimension 3, these singularities can be removed manually by a so-called surgery construction and the flow can be continued beyond them. The long-term goal of this project is to generalize this surgery construction to dimension 4, and possibly higher. To achieve this, the PI will study the singularity formation of higher dimensional Ricci flows, using a new theory he recently found. A successful construction in dimension 4 may have interesting topological and geometric applications. The PI will also further study Ricci flows with surgery, and the closely related "Ricci flows through singularities," in dimension 3 and find further geometric and topological applications. The award provides funds for graduate students to engage in research related to the project.The research project is split into two parts. The first project is a continuation of the PI's recently obtained compactness and partial regularity theory for Ricci flows in higher dimensions. The goal of the project is to use this new theory to construct a "Ricci flow with surgery'" or "Ricci flow through singularities" in dimension 4, generalizing the analogous construction in dimension 3. The strategy for achieving this is to deduce spatial asymptotic estimates on blow-up limits and use these to obtain a qualitative picture of the singularity formation. Based on this picture, the next step is to remove singularities via cylindrical and conical surgery constructions. The project also aims to characterize the long-time asymptotics of the flow. A successful construction and analysis may lead to several interesting topological and geometric applications. The second project continues work of the PI and collaborator on the uniqueness and continuous dependence of 3-dimensional singular Ricci flows through singularities. Previous work used this continuous dependence to resolve the Generalized Smale Conjecture (which classifies diffeomorphism groups of certain 3-manifolds up to homotopy) and a conjecture regarding the space of metrics with positive scalar curvature on 3-manifolds. This project will study more deeply the techniques used in these proofs, which have the potential to produce further results.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.