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

球体、平面(如一张平纸)和双曲曲面(如马鞍面)的行为各不相同。在球面上，从一点向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，或者至少不会比平坦表面上的移动速度快，而在马鞍面上，测地线曲线会迅速远离彼此。此行为与曲面的曲率有关，其中球体为正曲率，平面为零曲率，鞍部为负曲率。类似的行为也适用于自然界中出现的更高维度的物体。曲率在散热和散电等自然现象的行为中起着关键作用，对于不像上述三种类型那样平滑的物体也是如此。这种非光滑物体在自然界中存在，并且具有折痕、凹凸不平和类分形结构，因此经典的曲率理论不适用于它们。这个项目的重点是使用最近开发的针对非光滑环境的曲率模拟，并探索这种模拟如何决定这类对象中诸如散热等自然现象的行为。本项目的目标是探索度量空间的负曲率概念与非线性位势理论和拟共形映射之间的联系。所考虑的空间具有一致局部加倍测度，该测度支持一致局部Poincare不等式。光滑双曲流形是一个原型空间，它具有一个局部一致加倍测度，它支持一个一致局部Poincare不等式，但不支持它们的全局相似性质，经验告诉我们，这类空间在大尺度上具有指数体积增长。该项目分为三个部分。在该项目的第一部分，重点是空间的大尺度负曲率(如在最优质量运输中所给出的)及其与大尺度势能理论(非线性“热”能量耗散)和末端的双曲性的联系。在第二部分中，我们的目的是当空间具有Lott和Villani意义下有界的Ricci曲率时，构造连接空间中点对的曲线族。项目的第三部分是将有界二重非光滑空间作为Gromov-双曲填充的边界，并利用这一观点研究了弱路径连通空间上的非局部位势理论和非局部能量最小化的正则性。本文描述的研究构成了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，就像在Cheeger，Gromov和Perelman的作品中一样。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.