Ricci Flows and Steady Ricci Solitons

该奖项支持以利玛窦流为重点的微分几何研究。这些流是在带有度量的流形上定义的，也就是说，一种测量距离的方法。里奇流是黎曼度量的几何偏微分方程。利玛窦流倾向于将一个初始度量演变成一个更均匀的度量。里奇流的奇点分析是一个中心课题，因为它有助于理解流形的几何和拓扑结构。在这个方向上最显著的应用是佩雷尔曼对庞加莱猜想和几何化猜想的解析。许多里奇流奇点模型都是里奇孤子。最近由PI构建的孤子的例子看起来像飞翔的翅膀。PI将研究所有三维稳定Ricci孤子的几何，并尝试根据它们的渐近极限对它们进行分类。此外，PI将研究高维稳定里奇孤子，看看它们是否可以作为奇点模型出现。这个研究项目分为两个项目。第一个项目是证明所有三维稳定Ricci孤子的O(2)-对称性。这包括证明Bryant孤子是唯一的三维稳定Ricci孤子，它是渐近于射线的。这扩展了先前的PI结果，其中假设孤子具有O(2)对称性。PI开发了一些方法，这些方法可以扩展到更一般的3维古代崩塌里奇流。特别是，PI将在三维空间中研究古代崩塌的里奇流的对称性，并旨在通过某些二维极限对它们进行分类。第二个项目是PI关于非紧化初始流形的里奇流存在性理论的延续。PI将研究非紧致里奇流在拓扑和几何中的应用。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。

This award supports research in differential geometry focusing on Ricci flows. These flows are defined on manifolds equipped with a metric, that is to say, a way of measuring distance. A Ricci flow is a geometric partial differential equation for Riemannian metrics. The Ricci flow tends to evolve an initial metric into a more homogeneous one. The singularity analysis of Ricci flow is a central subject, as it helps to understand the geometry and topology of manifolds. The most remarkable application in this direction is the resolution of the Poincare conjecture and the Geometrization conjecture by Perelman. Many of the Ricci flow singularity models are Ricci solitons. Recent examples of solitons constructed by the PI look like flying wings. The PI will study the geometry of all 3-dimensional steady Ricci solitons and try to classify them by their asymptotic limits. In addition, the PI will study the higher-dimensional steady Ricci solitons and see if they can arise as singularity models.The research project is split into two projects. The first project is to prove the O(2)-symmetry of all 3-dimensional steady Ricci solitons. This includes showing that the Bryant soliton is the unique 3-dimensional steady Ricci soliton that is asymptotic to a ray. This extends a previous result of the PI in which one assumes the O(2)-symmetry of the soliton. The PI developed some methods that may be extended to the more general class of ancient collapsed Ricci flows in dimension 3. In particular, the PI will investigate the symmetry of the ancient collapsed Ricci flows in dimension 3 and aim at classifying them by certain 2-dimensional limits. The second project is a continuation of the PI's work on the existence theory of Ricci flows coming out of non-compact initial manifolds. The PI will investigate the applications of non-compact Ricci flows in topology and geometry.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.