Quasiconformal Extensions to Space of Weierstrass-Enneper Lifts

TL;DR

This paper develops a method to extend certain harmonic mappings from the unit disk to 3-space quasiconformally, using circle fibrations and convexity, generalizing classical results like Ahlfors-Weill.

Contribution

It introduces a new quasiconformal extension technique for Weierstrass-Enneper lifts of harmonic mappings, based on space fibrations and convexity considerations.

Findings

Provides explicit quasiconformal extension formulas

Establishes a sufficient condition for univalence of harmonic mappings

Generalizes the Ahlfors-Weill extension to 3-space

Abstract

We derive a quasiconformal extension to 3-space of the Weierstrass-Enneper lifts of a class of harmonic mappings defined in the unit disk. The extension is based on fibrations of space by circles in domain and image that correspond to each other in a natural way. Convexity plays an essential role in the analysis. As a corollary we derive a sufficient condition for the underlying harmonic mapping to be univalent in the disk, with an explicit quasiconformal extension to the extended plane that generalizes the well known formula by Ahlfors-Weill.