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Notions of Curvature and Their Role in Analysis on Metric Measure Spaces

曲率的概念及其在度量测度空间分析中的作用

基本信息

批准号：
1800161
负责人：
Nageswari Shanmugalingam
金额：
$ 24万
依托单位：
University of Cincinnati Main Campus
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800161&HistoricalAwards=false
关键词：
Notions Curvature Their Role Analysis

项目摘要

The sphere, a flat surface such as a flat piece of paper, and a hyperbolic surface such as a saddle behave differently from each other. On the surface of the ball curves of least length (geodesics) emanating from a point in different directions tend to not move away from each other in the short term, or at least less rapidly than in the flat surface, whereas in the saddle surface the geodesics curve away from each other rapidly. This behavior is related to the curvature of the surface, with the sphere of positive curvature, the flat surface of zero curvature, and the saddle of negative curvature. Analogs of this behavior hold in higher dimensional objects that occur in nature. Curvature plays a key role in how natural phenomena such as dissipation of heat and electricity behave, and this is true also in objects that are not as smooth as the three types described above. Such non-smooth objects occur in nature and have creases, bumps and fractal-like structure, and so the classical theory of curvature does not apply to them. The focus of this project is to use the analog of curvature developed for the non-smooth setting recently, and explore how that dictates the behavior of natural phenomena such as heat dissipation in such objects.The goal of this project is to explore links between the notions of negative curvature of a metric space on the one hand, and nonlinear potential theory and quasiconformal mappings on the other hand. The spaces considered are equipped with a uniformly locally doubling measure supporting a uniformly local Poincare inequality. A prototype space equipped with a uniformly locally doubling measure supporting a uniformly local Poincare inequality but does not support their global analogs is the smooth hyperbolic manifold, and experience tells us that such spaces have exponential volume growth at large scales. This project is divided into three parts. In the first part of the project, the focus is the large scale negative curvature of the space (as given in optimal mass transportation) and its connections to large-scale potential theory (non-linear ``heat" energy dissipation) and hyperbolicity of ends. In the second part the aim is to construct geometric families of curves connecting pairs of points in the space when the space has lower bounded Ricci curvature in the sense of Lott and Villani. The third part of the project is to consider bounded doubling nonsmooth spaces as boundaries of Gromov-hyperbolic filling and use this perspective to study non-local potential theory and regularity of nonlocal energy minimizers on poorly pathconnected spaces. The research described herein forms a part of the program of quasiconformal classification of nonsmooth spaces. Such spaces arise in the study of smooth manifolds when considering Gromov-Hausdorff limit spaces as in the works of Cheeger, Gromov, and Perelman.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.

球面、平面（如一张平面纸）和双曲面（如马鞍）的行为彼此不同。在球面上，从一点沿不同方向发出的最短长度的曲线（测地线）在短期内不会彼此远离，或者至少不如在平面上那样快，而在鞍面上，测地线会迅速彼此远离。这种行为与表面的曲率有关，球面为正曲率，平坦表面为零曲率，马鞍为负曲率。这种行为的类似物存在于自然界中发生的更高维度的物体中。曲率在自然现象中起着关键作用，例如热量和电力的耗散，这在不像上面描述的三种类型那样光滑的物体中也是如此。这种非光滑物体在自然界中存在，具有折痕，凸起和分形结构，因此经典的曲率理论不适用于它们。本课题的研究重点是利用近年来在非光滑环境下发展起来的曲率模拟，探索其如何支配热耗散等自然现象的行为。本课题的目的是探索度量空间的负曲率概念与非线性势理论、拟共形映射之间的联系。所考虑的空间配备了一个一致局部加倍措施支持一致局部庞加莱不等式。一个原型空间配备了一个一致局部加倍测度，支持一个一致局部庞加莱不等式，但不支持它们的整体类似物，这是光滑双曲流形，经验告诉我们，这样的空间在大尺度上有指数体积增长。本项目分为三个部分。在该项目的第一部分中，重点是空间的大规模负曲率（如最佳质量运输中所给出的）及其与大规模势理论（非线性“热”能量耗散）和双曲性的联系。结束。在第二部分中，我们的目的是建立几何家庭的曲线连接点对的空间时，空间有下界Ricci曲率的意义下，洛特和Villani。第三部分是考虑有界加倍非光滑空间作为Gromov双曲填充的边界，并利用这一观点研究非局部势理论和非局部能量极小点在不良路径连通空间上的正则性。本文所描述的研究是非光滑空间拟共形分类程序的一部分。这样的空间出现在光滑流形的研究中，当考虑Gromov-Hausdorff极限空间时，如Cheeger，Gromov和Perelman的作品。该奖项反映了NSF的法定使命，并被认为值得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来支持。