Hopf differentials and smoothing Sobolev homeomorphisms

TL;DR

This paper demonstrates that planar homeomorphisms can be approximated by smooth diffeomorphisms in Sobolev spaces and applies this to show that certain mappings with holomorphic Hopf differentials are harmonic.

Contribution

It establishes approximation results for Sobolev homeomorphisms and characterizes mappings with holomorphic Hopf differentials as harmonic.

Findings

Planar homeomorphisms can be approximated by diffeomorphisms in W^{1,2}

Mappings with holomorphic Hopf differentials are harmonic

Approximation holds in the Royden algebra

Abstract

We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space $W^{1,2}$ and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.