Quasiconformal extensions, Loewner chains, and the lambda-Lemma

TL;DR

This paper provides a new proof and generalization of Becker's 1972 criterion for quasiconformal extendibility of Loewner chains, characterizes the resulting extensions, and explores their extremal properties using the lambda-Lemma.

Contribution

It introduces a novel proof based on the lambda-Lemma, characterizes all Becker-type extensions, and analyzes their extremality in the context of quasiconformal mappings.

Findings

New proof of Becker's quasiconformal extension criterion

Complete characterization of Becker's extension constructions

Examples of extremal and non-extremal Becker extensions

Abstract

In 1972, J. Becker [J. Reine Angew. Math. 255] discovered a sufficient condition for quasiconformal extendibility of Loewner chains. Many known conditions for quasiconformal extendibility of holomorphic functions in the unit disk can be deduced from his result. We give a new proof of (a generalization of) Becker's result based on Slodkowski's Extended lambda-Lemma. Moreover, we characterize all quasiconformal extensions produced by Becker's (classical) construction and use that to obtain examples in which Becker's extension is extremal (i.e. optimal in the sense of maximal dilatation) or, on the contrary, fails to be extremal.