Definitions of quasiconformality on metric surfaces

TL;DR

This paper investigates various definitions of quasiconformality on metric surfaces, establishing their equivalence and leading to a new uniformization theorem for such surfaces.

Contribution

It proves the equivalence of different quasiconformality notions on metric surfaces and introduces a novel uniformization result.

Findings

Finite distortion implies maximal and minimal stretchings are finite.

Equivalence of multiple quasiconformality definitions.

New uniformization theorem for metric surfaces.

Abstract

We explore the interplay between different definitions of distortion for mappings $f\colon X\to \mathbb{R}^2$, where $X$ is any metric surface, meaning that $X$ is homeomorphic to a domain in $\mathbb{R}^2$ and has locally finite 2-dimensional Hausdorff measure. We establish that finite distortion in terms of the familiar analytic definition always implies finite distortion in terms of maximal and minimal stretchings along paths. The converse holds for maps with locally integrable distortion. In particular, we prove the equivalence of various notions of quasiconformality, implying a novel uniformization result for metric surfaces.