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

Abstract

If \(\Omega\) is a simply connected domain in \(\overline{\mathbf C}\) then, according to the Ahlfors-Gehring theorem, \(\Omega\) is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in \(\Omega\) 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 \(\overline{\mathbf C}\) and, under the additional assumption of quasiconformality in \(\Omega\), they admit a quasiconformal extension to \(\overline{\mathbf C}\). The Ahlfors-Gehring theorem has been extended to finitely connected domains \(\Omega\) by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in \(\Omega\) if and only if the components of \(\partial\Omega\) are either points or quasicircles. We generalize this theorem to harmonic mappings.