Quasiconformal harmonic mappings between two doubly connected domains in the plane

TL;DR

This paper proves the existence of a quasiconformal harmonic diffeomorphism between two doubly-connected domains (annuli) with a controlled distortion factor depending on their moduli ratio, extending known energy-minimal mappings.

Contribution

It provides a concise proof that such quasiconformal harmonic diffeomorphisms exist for annuli with moduli ratio at most one, with the distortion approaching one as the ratio approaches one.

Findings

Existence of a $K$-quasiconformal harmonic diffeomorphism for annuli with $ ext{Mod}( ext{domain}) \\le ext{Mod}( ext{target})$

The distortion $K$ depends on the moduli ratio and tends to 1 as the ratio approaches 1

Simplified proof of the existence of energy-minimal harmonic diffeomorphisms in doubly-connected domains.

Abstract

It is known for some time that there exists an energy-minimal diffeomorphism between two doubly-connected domains $\Omega$ and $D$ provided that $\mathrm{Mod}(\Omega)\le \mathrm{Mod}{D}$ and that diffeomorphism is harmonic \cite{tedi}. In this note we give a short proof of the fact that for given annuli $\Omega$ and $D$ satisfying the condition $\mathrm{Mod}(\Omega)\le \mathrm{Mod}(D)$ there exist a $K-$quasiconformal harmonic diffeomorphism $f:\Omega\onto D$, where $K=K(\tau)$, $\tau=\mathrm{Mod}(\Omega)/ \mathrm{Mod}(D)$ and $\lim_{\tau\to 1}K(\tau)=1$.