Monotone Sobolev extensions in metric surfaces and applications to uniformization

TL;DR

This paper establishes a monotone Sobolev extension theorem for maps into Jordan domains in metric surfaces and applies it to achieve a uniformization result for compact metric surfaces through energy minimization.

Contribution

It introduces a new monotone Sobolev extension theorem for metric surfaces and uses it to prove a uniformization theorem for compact metric surfaces.

Findings

Proved a monotone Sobolev extension theorem for metric surfaces.

Achieved a uniformization result for compact metric surfaces.

Demonstrated energy minimization in the class of monotone Sobolev maps.

Abstract

We prove a monotone Sobolev extension theorem for maps to Jordan domains with rectifiable boundary in metric surfaces of locally finite Hausdorff 2-measure. This is then used to prove a uniformization result for compact metric surfaces by minimizing energy in the class of monotone Sobolev maps.