Ricci Curvature, Ricci Flow and Foliations

这个建议涉及到高维的弯曲空间。一个有趣的问题是如何测量非光滑空间的曲率。在截面曲率的情况下，这是由亚历山大·S在1930年提出的，他给出了奇异空间具有非负截面曲率意味着什么的很好的概念。最近，提出者与塞德里克·维拉尼以及卡尔-西奥多·斯图姆的相关工作给出了对于具有非负Ricci曲率的奇异空间意味着什么的很好的概念。所提出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，具有以下有界的Ricci曲率的空间的性质。在不同的方向上，Ricci曲率提供了一种通过Ricci流来平滑空间几何形状的方法。这一流是由理查德·汉密尔顿在20世纪80年代由S引入的，他用它刻画了具有非负Ricci曲率的三维光滑空间的拓扑。最近，Perelman利用Ricci流证明了三维拓扑学中最大的猜想，即Poincare猜想和瑟斯顿几何化猜想。尽管佩雷尔曼取得了巨大的成就，但关于利玛窦在三维和更高维度的流动，仍有许多悬而未决的问题。这项拟议的研究将解决其中的一些问题。另一种奇异空间的产生方式是当高维空间被分成低维空间时。这种叶状结构的参数化空间几乎总是拓扑奇性的，本文研究的部分内容是对这类空间进行分析，或者更准确地说，证明一个横向指数定理。对于高维光滑空间，不一定居住在平坦空间中，这些不同的概念称为截面曲率、Ricci曲率和标量曲率。每一次都是前一次的平均，即Ricci曲率是截面曲率的平均，标量曲率是Ricci曲率的平均。利玛窦的曲率是通过爱因斯坦的广义相对论方程进入物理学的。上述工作的一部分涉及“最佳输运”。这是对在弯曲空间中传输质量的最佳方式的研究。它是由蒙格在17世纪80年代由S提出的，近几十年来随着偏微分方程组和应用数学的应用而复兴。我们已经证明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。

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.