Ricci Flows and Steady Ricci Solitons

该奖项支持微分几何的研究重点是里奇流。这些流动是在配备了公制的流形上定义的，也就是说，一种测量距离的方法。Ricci流是黎曼度量的几何偏微分方程。Ricci流倾向于将初始度量演化为更齐次的度量。里奇流的奇异性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。在这个方向上最显著的应用是Perelman对庞加莱猜想和几何化猜想的解决。许多Ricci流奇异性模型都是Ricci孤子。最近由PI构造的孤子的例子看起来像飞行的翅膀。PI将研究所有三维稳态Ricci孤子的几何形状，并试图根据它们的渐近极限对它们进行分类。此外，PI将研究高维稳态Ricci孤子，看看它们是否可以作为奇异模型出现。第一个方案是证明所有三维定常Ricci孤子的O（2）对称性。这包括证明布莱恩特孤立子是唯一的3维稳定里奇孤立子，它对射线是渐进的。这推广了PI中假设孤子具有O（2）对称性的一个结果。PI开发了一些方法，可以扩展到更一般的类古代崩溃的里奇流在3维。特别是，PI将研究三维中古代坍缩的Ricci流的对称性，并旨在通过某些二维限制对其进行分类。第二个项目是继续PI的工作存在理论的Ricci流出来的非紧初始流形。PI将研究非紧致Ricci流在拓扑学和几何学中的应用。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。

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.