Quasiconformal extension for harmonic mappings on finitely connected domains

TL;DR

This paper establishes conditions under which harmonic quasiconformal mappings on finitely connected domains can be extended to the entire plane, highlighting the role of boundary components and Schwarzian derivative size.

Contribution

It proves that harmonic quasiconformal mappings with small Schwarzian derivative extend to the plane on domains with point or quasicircle boundaries, and discusses univalence criteria on uniform domains.

Findings

Harmonic quasiconformal mappings extend to the plane if Schwarzian derivative is small.

Extension is possible on domains with boundary components as points or quasicircles.

Univalence criterion applies on uniform domains.

Abstract

We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a univalence criterion for harmonic mappings holds on uniform domains.