Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains - Efraimidis, Iason (Annales Fennici Mathematici, 2021)
If \(Ω\) is a simply connected domain in \(C\) then, according to the Ahlfors-Gehring theorem, \(Ω\) is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in \(Ω\) 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 \(C\) and, under the additional assumption of quasiconformality in \(Ω\), they admit a quasiconformal extension to \(C\). The Ahlfors-Gehring theorem has been extended to finitely connected domains \(Ω\) by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in \(Ω\) if and only if the components of \(Ω\) are either points or quasicircles. We generalize this theorem to harmonic mappings.