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arXiv · 1005.4937

Ahlfors-Weill Extensions for a Class of Minimal Surfaces

Martin Chuaqui·Peter Duren·Brad Osgood
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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.

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Martin Chuaqui, Peter Duren, Brad Osgood. 2010-05-26. Ahlfors-Weill Extensions for a Class of Minimal Surfaces. https://arxiv.org/abs/1005.4937

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