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.

该奖项支持针对Ricci流的差异几何形状的研究。这些流是在装有度量标准的歧管上定义的，也就是说，是一种测量距离的方法。 RICCI流是Riemannian指标的几何部分微分方程。 RICCI流倾向于将初始度量发展为更均匀的度量。 RICCI流的奇异性分析是一个中心主题，因为它有助于理解歧管的几何形状和拓扑。朝这个方向上的最显着应用是庞塞雷猜想的解决和佩雷尔曼的几何化猜想。 RICCI流动奇点模型是Ricci孤子。 PI构建的唯一的唯一例子看起来像飞翅。 PI将研究所有三维稳定RICCI孤子的几何形状，并试图通过其渐近极限对其进行分类。此外，PI将研究高维稳定的RICCI孤子，看看是否可以作为奇异模型出现。研究项目分为两个项目。第一个项目是证明所有3维稳定RICCI孤子的O（2） - 对称性。这包括表明科比孤子是渐近射线的独特的3维稳定Ricci Soliton。这扩展了PI的先前结果，其中一个假定孤子的O（2） - 对称性。 PI开发了一些方法，这些方法可能会扩展到更古老的倒塌的Ricci流中的更一般类别。尤其是，PI将研究尺寸3中古代折叠的Ricci流的对称性，并旨在通过某些2维限制对它们进行分类。第二个项目是PI对Ricci流的存在理论的延续，该理论是由非紧凑型初始流形出来的。 PI将调查非紧缩RICCI流在拓扑和几何学中的应用。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来支持的。