Uniformization of metric surfaces: A survey

TL;DR

This survey reviews recent advances in the uniformization of metric surfaces, including classical theorems and modern results on quasisymmetric and quasiconformal uniformization of various metric surfaces.

Contribution

It summarizes key recent theorems on uniformization of metric surfaces, highlighting developments from classical to fractal and arbitrary surfaces.

Findings

Bonk-Kleiner theorem on quasisymmetric uniformization of metric spheres

Rajala's quasiconformal uniformization of metric spheres

Romney and author's work on uniformization of arbitrary metric surfaces

Abstract

In this survey we present the most recent developments in the uniformization of metric surfaces, i.e., metric spaces homeomorphic to two-dimensional topological manifolds. We start from the classical conformal uniformization theorem of Koebe and Poincar\'e. Then we discuss the Bonk-Kleiner theorem on the quasisymmetric uniformization of metric spheres, which marks the beginning of the study of the uniformization problem on fractal surfaces. The next result presented is Rajala's theorem on the quasiconformal uniformization of metric spheres. We conclude with the final result in this series of works, due to Romney and the author, on the weakly quasiconformal uniformization of arbitrary metric surfaces of locally finite area under no further assumption.