Uniformization Of Metric Surfaces Using Isothermal Coordinates

TL;DR

This paper proves that certain metric surfaces can be transformed into Riemannian surfaces using isothermal coordinates, extending classical uniformization results to more general metric spaces.

Contribution

It introduces a method to construct isothermal coordinates for metric surfaces, establishing a uniformization theorem in the setting of metric geometry.

Findings

Metric surfaces covered by quasiconformal images of Euclidean domains are quasiconformally equivalent to Riemannian surfaces.

Construction of isothermal coordinates for metric surfaces.

Extension of classical uniformization to metric spaces with finite Hausdorff measure.

Abstract

We establish a uniformization result for metric surfaces - metric spaces that are topological surfaces with locally finite Hausdorff 2-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct suitable isothermal coordinates.