Ahlfors-Weill Extensions for a Class of Minimal Surfaces

TL;DR

This paper generalizes the Ahlfors-Weill extension to harmonic mappings lifting to minimal surfaces, creating homeomorphic, quasiconformal extensions via boundary reflections, and broadening the geometric understanding of minimal surface extensions.

Contribution

It introduces a novel extension method for harmonic mappings to minimal surfaces, using boundary reflection and hyperbolic convexity, extending the classical Ahlfors-Weill formula.

Findings

Produces homeomorphic, quasiconformal extensions of minimal surfaces

Utilizes boundary reflection across Euclidean circles orthogonal to the surface

Extends the minimal surface with real-analytic properties off the boundary

Abstract

The Ahlfors-Weill extension of a conformal mapping of the disk is generalized to the lift of a harmonic mapping of the disk to a minimal surface, producing homeomorphic and quasiconformal extensions. The extension is obtained by a reflection across the boundary of the surface using a family of Euclidean circles orthogonal to the surface. This gives a geometric generalization of the Ahlfors-Weill formula and extends the minimal surface. Thus one obtains a homeomorphism of $\overline{\mathbb{C}}$ onto a toplological sphere in $\overline{\mathbb{R}^3} = \mathbb{R}^3 \cup \{\infty\}$ that is real-analytic off the boundary. The hypotheses involve bounds on a generalized Schwarzian derivative for harmonic mappings in term of the hyperbolic metric of the disk and the Gaussian curvature of the minimal surface. Hyperbolic convexity plays a crucial role.