Metric geometry and functions of bounded variation

University Of Cincinnati Main Campus, Cincinnati OH

Investigators

Abstract

Objects occurring in nature rarely are smooth in appearance. From fractal objects to porous media, the equations that govern dissipation of quantities such as heat and pressure have non-smooth components. Such non-smooth objects also occur as limits of certain smooth objects. To understand and exploit the behavior of such objects, we need to remove the smoothness assumptions from Riemannian geometry. For such non-smooth objects we need to study behaviors of non-smooth energy minimizers and their connection to the geometry, that is, the structural properties, of the underlying space or object. This project on metric geometry and functions on bounded variation contributes to this goal by exploring interconnections between the geometry of the underlying metric space equipped with a measure, and the properties of sets of minimal surface areas. The study of objects here are metric spaces equipped with a measure that is doubling (that is, measure of a ball is comparable to the measure of a concentric ball of double the radius) and such that the variance of a Lipschitz continuous function on a ball can be controlled in terms of the average value of its energy (computed using its local oscillation) on that ball. In this setting, this project aims to study properties such as rectifiability, porosity, shape, and natural dimension of sets of locally minimal surface areas. Analysis on metric measure spaces arose from many sources; the study of complex analytic functions, differential equations governing fractures in mixed material and associated mappings of finite distortion, control theory in engineering and associated Carnot-Caratheodory spaces, the study of the famous Poincare conjecture, are some of the roots of this field of study. However, unlike in the Euclidean situation where structural properties of minimal surfaces (surfaces with smallest surface energy) are well understood, in the non-smooth setting that arise in nature and in control theory problems of physics and engineering, the structure of minimal surfaces is poorly understood. This project seeks to explore such structures and expand our knowledge of minimal surfaces in a non-smooth setting. The results of this study will be useful in further understanding the change in the behavior of objects that are transformed due to natural effects such as heat and pressure. In addition to advancing our knowledge of non-smooth objects, the study conducted in this project will also contribute to the ongoing development of the theory of analysis on metric spaces; a great benefit of this developing theory is that since much of the structural tools available in classical Euclidean analysis are not available in the non-smooth setting, new tools and methods have necessarily to be developed, making the theory more accessible to a wider audience.