Ricci Curvature, Ricci Flow and Foliations

里奇曲率、里奇流和叶状结构

• 批准号: 0903076
• 负责人: John Lott
• 金额: $ 32.99万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0903076&HistoricalAwards=false
• 关键词: Ricci; Curvature; Flow; Foliations

This proposal deals with curved spaces of higher dimension.An interesting question is how to measure the curvature of a nonsmooth space. In the case of sectional curvature, this was initiated in the 1930's by Alexandrov, who gave a good notion of what it means for a singular space to have nonnegative sectional curvature.Recent work by the proposer, joint with Cedric Villani, together with related work by Karl-Theodor Sturm, has given a good notion of what it means for a singular space to have nonnegative Ricci curvature. The proposed research will explore properties of spaces with nonnegative Ricci curvature or, more generally, with Ricci curvature bounded below. In a different direction, the Ricci curvature gives a way to smooth out the geometry of a space, by means of the Ricci flow. This flow was introduced by Richard Hamilton in the 1980's, who used it to characterize the topology of three-dimensional smooth spaces with nonnegative Ricci curvature. Recently, Perelman has proved the biggest conjectures in three-dimensional topology, namely the Poincare Conjecture and Thurston's Geometrization Conjecture, using Ricci flow. Despite Perelman's great achievements, there are many open questions concerning the Ricci flow in dimension three and in higher dimensions. The proposed research will address some of these questions.Another way that singular spaces arise is when a higher-dimensional space is foliated into lower dimensional spaces. The parametrizing space for such a foliation is almost always topologically singular.Part of the proposed research is to do analysis on such spaces, or more precisely to prove a transverse index theorem.There arevarious notions of curvature, which coincide in the traditional setting of two-dimensional surfaces in three-dimensional space. For a higher dimensional smooth space, not necessarily living in a flat space, these different notions are called the sectional curvature, the Ricci curvature and the scalar curvature. Each one is an averaging of the previous one, i.e. the Ricci curvature is an averaging of sectional curvature and the scalar curvature is an averaging of Ricci curvature. The Ricci curvature enters in physics through Einstein's equations of general relativity.Part of the work described above concerns ``optimal transport''. This is the study of the optimal way to transport mass in a curved space. It was initiated by Monge in the 1780's and has had a revival in recent decades, with application to partial differential equations and applied mathematics.We have shown that it also has application in differential geometry. Conversely, concepts from differential geometry have application to optimal transport.

这一建议涉及高维的弯曲空间，一个有趣的问题是如何度量非光滑空间的曲率。 在截面曲率的情况下，这是由亚历山德罗夫在20世纪30年代提出的，他给出了一个很好的概念，这意味着一个奇异的空间有非负截面曲率。最近的工作由提议者，联合Cedric Villani，连同相关的工作卡尔-西奥多Sturm，给出了一个很好的概念，这意味着一个奇异的空间有非负里奇曲率。 拟议的研究将探索具有非负Ricci曲率的空间的性质，或者更一般地说，具有Ricci曲率有界的空间。 在另一个方向上，里奇曲率通过里奇流提供了一种平滑空间几何的方法。 这个流是由理查德汉密尔顿在20世纪80年代提出的，他用它来刻画具有非负里奇曲率的三维光滑空间的拓扑。 最近Perelman利用Ricci流证明了三维拓扑中最大的猜想，即Poincare猜想和Thurston的几何化猜想。 尽管佩雷尔曼的伟大成就，有许多开放的问题，利玛窦流在三维和更高的维度。 奇异空间的另一种产生方式是当一个高维空间被分割成低维空间时。 这种叶理的参数化空间几乎都是拓扑奇异的，部分研究工作是对这种空间进行分析，或者更准确地说，是证明一个横截指标定理，有各种各样的曲率概念，它们在传统的三维空间中的二维曲面的设置中是一致的。 对于高维光滑空间，不一定是平坦空间，这些不同的概念被称为截面曲率，Ricci曲率和标量曲率。 每一个都是前一个的平均，即里奇曲率是截面曲率的平均，标量曲率是里奇曲率的平均。里奇曲率通过爱因斯坦的广义相对论方程进入物理学。上述工作的一部分涉及“最优传输”。 这是研究在弯曲空间中运输质量的最佳方式。它是由Monge在1780年提出的，近几十年来又得到了复兴，并应用于偏微分方程和应用数学。我们已经证明了它在微分几何中也有应用。 相反，微分几何的概念也适用于最优运输。