Canonical parametrizations of metric surfaces of higher topology

TL;DR

This paper provides an alternative proof that certain metric surfaces with higher topology are quasisymmetrically equivalent to model surfaces, extending uniformization results to surfaces with boundary and canonical parametrizations.

Contribution

It offers a new proof of a generalized uniformization theorem for metric surfaces, including those with boundary, using local quasisymmetric parametrizations.

Findings

Metric surfaces are quasisymmetrically equivalent to model surfaces of the same topology.

The equivalence extends to surfaces with non-empty boundary.

The associated maps can be chosen in a canonical way.

Abstract

We give an alternate proof to the following generalization of the uniformization theorem by Bonk and Kleiner. Any linearly locally connected and Ahlfors 2-regular closed metric surface is quasisymmetrically equivalent to a model surface of the same topology. Moreover, we show that this is also true for surfaces as above with non-empty boundary and that the corresponding map can be chosen in a canonical way. Our proof is based on a local argument involving the existence of quasisymmetric parametrizations for metric discs as shown in in a paper of Lytchak and Wenger.