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

University Of Cincinnati Main Campus, Cincinnati OH

Investigators

Abstract

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.