Ricci Flow

Ricci流是一种几何过程，可以用来改进给定的几何形状，使其更均匀。Ricci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维中的广义Smale猜想。一般的期望是，里奇流在极限中产生一种几何，这种几何在某种意义上是拓扑所固有的，也就是说，底层空间的松散构成。然而，通常Ricci流在有限时间内会产生复杂的奇异性。在三维空间中，这些奇点可以通过所谓的外科手术构造手动移除，并且流动可以继续超过它们。这个项目的长期目标是将这种手术构造推广到4维，甚至更高。为了实现这一目标，PI将研究更高维的里奇流的奇点形成，使用他最近发现的一个新理论。在4维成功的建设可能有有趣的拓扑和几何应用。PI还将进一步研究手术的Ricci流，以及密切相关的“Ricci流通过奇点”，在3维中，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。研究项目分为两个部分。第一个项目是继续PI最近获得的紧致性和部分正则性理论的Ricci流在更高的维度。该项目的目标是使用这个新的理论来构建一个四维的“带手术的里奇流”或“通过奇点的里奇流”，推广三维的类似结构。实现这一目标的策略是推导出空间渐近估计爆破限制，并使用这些来获得一个定性的奇异性形成的图片。基于这张图片，下一步是通过圆柱形和圆锥形手术结构来去除奇点。该项目还旨在描述流的长时间渐近特性。一个成功的构造和分析可能会导致几个有趣的拓扑和几何应用。第二个项目继续工作的PI和合作者的独特性和连续依赖的三维奇异里奇流通过奇点。以前的工作使用这种连续的依赖来解决广义Smale猜想（它分类的同伦某些3流形的同胚群）和一个猜想关于空间的度量与正标量曲率的3流形。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NSF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.