Quantitative quasisymmetric uniformization of compact surfaces

TL;DR

This paper extends Bonk and Kleiner's result, showing that any compact orientable surface with certain metric properties can be quantitatively uniformized to a standard surface, broadening the scope of quasisymmetric uniformization.

Contribution

It generalizes the quasisymmetric uniformization from spheres to all compact orientable surfaces with Ahlfors regularity and local contractibility.

Findings

Extended uniformization to all compact orientable surfaces.

Provided quantitative bounds for the uniformization.

Broadened applicability of quasisymmetric equivalence.

Abstract

Bonk and Kleiner showed that any metric sphere which is Ahlfors 2-regular and linearly locally contractible is quasisymmetrically equivalent to the standard sphere, in a quantitative way. We extend this result to arbitrary metric compact orientable surfaces.