Monotone Sobolev extensions in metric surfaces and applications to uniformization

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.

arXiv commentSimplified proof of Lemma 3.2 and added funding source