Quasiconformal uniformization of metric surfaces

Abstract

The main subject of this dissertation is the uniformization problem for nonsmooth surfaces. The foundational question is to find necessary and sufficient conditions for the existence of a homeomorphism taking a given nonsmooth surface into a smooth Riemannian surface while requiring minimal geometric distortion from the mapping. More specifically, we require the homeomorphism to be quasiconformal. Our approach is based on a recent work by Rajala. The dissertation consists of four articles. In article [A], we prove a uniformization result for every nonsmooth surface satisfying mild geometric assumptions. In fact, we only assume that the surface can be covered by domains which can be quasiconformally mapped into the Euclidean plane. We prove that this is a sufficient (and necessary) condition for there to exist a quasiconformal map on to a smooth Riemannian surface. In article [B], the author and Romney investigate weighted distances on the Euclidean plane. The main result of the article shows a surprising link between the nonsmooth uniformization problem and sets removable for conformal mappings, a notion of removability introduced by Ahlfors and Beurling in the1950s. In article [C], we examine the boundary structure of nonsmooth Euclidean disks which have finite two-dimensional Hausdorff measure and whose interiors can be quasiconformally mapped on to the Euclidean disk. We prove a generalized Carathéodory theorem in this setting and provide examples showing the sharpness of the result. In article [D], we consider a metric version of the classical welding problem from complex analysis. We construct nonsmooth spheres by metrically welding the southern and northern hemispheres of the two-dimensional sphere along the equator using a homeomorphism from the equator onto itself. The goal is to understand when the resulting sphere can be quasiconformally mapped to the Euclidean sphere. A necessary condition we establish connects the metric welding problem to the classic alone, while our sufficient conditions are related to measure-theoretic properties and modulus of continuity of the welding map.