Quasiconformal extension for harmonic mappings on finitely connected domains

Iason Efraimidis

doi:10.5802/crmath.233

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.

Highlights

Let f be a harmonic mapping in a planar domain D and let ω = fz / fz be its dilatation
The Schwarzian derivative of f was defined by Hernández and Martín [9] as
When f is holomorphic this reduces to the classical Schwarzian derivative

Summary

Introduction

According to a theorem of Ahlfors [1], if D is a K -quasidisk there exists a constant c > 0, depending only on K , such that if f is analytic in D with S f D ≤ c f is univalent in D and has a quasiconformal extension to C This has been generalized by Osgood [12] to the case when D is a finitely connected domain whose boundary components are either points or quasicircles. For harmonic mappings and the definition (1) of the Schwarzian derivative, a univalence and quasiconformal extension criterion in the unit disk D was proved by Hernández and Martín [8] This was recently generalized to quasidisks by the present author in [5].

Quasiconformal Decomposition

Proof of Theorem 1

Remarks on uniform domains