Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains

Abstract

If $\mathrm{\Omega }$ is a simply connected domain in $\bar{\mathbf{C}}$ then, according to the Ahlfors-Gehring theorem, $\mathrm{\Omega }$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $\mathrm{\Omega }$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\bar{\mathbf{C}}$ and, under the additional assumption of quasiconformality in $\mathrm{\Omega }$, they admit a quasiconformal extension to $\bar{\mathbf{C}}$. The Ahlfors-Gehring theorem has been extended to finitely connected domains $\mathrm{\Omega }$ by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in $\mathrm{\Omega }$ if and only if the components of $\partial \mathrm{\Omega }$ are either points or quasicircles. We generalize this theorem to harmonic mappings.