Morrey's $\varepsilon$-conformality lemma in metric spaces

TL;DR

This paper offers a simplified proof and an extension of Morrey's lemma on epsilon-conformal mappings, applicable to Sobolev maps in metric spaces, with implications for minimal surface existence.

Contribution

It provides a more accessible proof of Morrey's lemma that extends to Sobolev maps into metric spaces, avoiding reliance on the measurable Riemann mapping theorem.

Findings

Simplified proof of Morrey's epsilon-conformality lemma

Extension to Sobolev maps in metric spaces

Applications to existence of higher genus area-minimizing surfaces

Abstract

We provide a simpler proof and slight strengthening of Morrey's famous lemma on $\varepsilon$-conformal mappings. Our result more generally applies to Sobolev maps with values in a complete metric space and we obtain applications to the existence of area minimizing surfaces of higher genus in metric spaces. Unlike Morrey's proof, which relies on the measurable Riemann mapping theorem, we only need the existence of smooth isothermal coordinates established by Korn and Lichtenstein.