Ricci Flows and Steady Ricci Solitons

里奇流和稳态里奇孤子

• 批准号: 2203310
• 负责人: Yi Lai
• 金额: $ 17.39万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203310&HistoricalAwards=false
• 关键词: Ricci; Flows; Steady; Solitons

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.

该奖项支持以 Ricci 流为重点的微分几何研究。这些流量是在配备有度量的流形上定义的，也就是说，是一种测量距离的方式。里奇流是黎曼度量的几何偏微分方程。 Ricci 流倾向于将初始度量发展为更均匀的度量。里奇流的奇异性分析是一个中心主题，因为它有助于理解流形的几何和拓扑。在这个方向上最引人注目的应用是解决庞加莱猜想和佩雷尔曼的几何化猜想。许多 Ricci 流奇点模型都是 Ricci 孤子。最近由 PI 构建的孤子例子看起来就像飞翔的翅膀。 PI 将研究所有 3 维稳态 Ricci 孤子的几何形状，并尝试根据渐近极限对它们进行分类。此外，PI还将研究高维稳态Ricci孤子，看看它们是否可以作为奇点模型出现。该研究项目分为两个项目。第一个项目是证明所有 3 维稳态 Ricci 孤子的 O(2) 对称性。这包括证明布莱恩特孤子是渐近于射线的唯一 3 维稳定里奇孤子。这扩展了 PI 的先前结果，其中假设孤子具有 O(2) 对称性。 PI 开发了一些方法，可以扩展到更一般的 3 维古代塌缩 Ricci 流类别。特别是，PI 将研究 3 维古代塌陷 Ricci 流的对称性，并旨在通过某些 2- 对它们进行分类。尺寸限制。第二个项目是 PI 关于来自非紧初始流形的 Ricci 流存在理论工作的延续。 PI 将研究非紧 Ricci 流在拓扑和几何中的应用。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。