Quasiconformal uniformization of metric surfaces of higher topology

TL;DR

This paper proves a uniformization theorem for metric surfaces of higher topology, establishing conditions under which they admit quasiconformal parametrizations, extending classical results to more general metric spaces.

Contribution

It introduces new conditions for metric surfaces to admit quasiconformal parametrizations, extending uniformization results to higher topology surfaces with minimal assumptions.

Findings

Existence of quasiconformal almost parametrizations for certain metric surfaces.

Conditions under which these parametrizations upgrade to quasisymmetries.

Extension of classical uniformization theorems to higher topology metric surfaces.

Abstract

We establish the following uniformization result for metric spaces $X$ of finite Hausdorff 2-measure. If $X$ is homeomorphic to a smooth 2-manifold $M$ with non-empty boundary, then we show that $X$ admits a quasiconformal almost parametrization $M\to X$, by only assuming that $X$ is locally geodesic and has rectifiable boundary. In particular, we recover a corollary of Ntalampekos and Romney by using the solution of the Plateau problem. After putting additional assumptions on $X$, we show that the quasiconformal almost parametrization upgrades to a quasisymmetry or a geometrically quasiconformal map, implying statements analogous to the uniformization theorems of Bonk and Kleiner as well as Rajala for surfaces of higher topology.