To the memory of Professor F. W. Gehring
Abstract
The Ahlfors-Weill extension of a conformal mapping of the disk is generalized to the Weierstrass-Enneper lift of a harmonic mapping of the disk to a minimal surface, producing homeomorphic and quasiconformal extensions to space. The extension is defined through the family of best Möbius approximations to the lift applied to a bundle of euclidean circles orthogonal to the disk. Extension of the planar harmonic map is also obtained subject to additional assumptions on the dilatation. The hypotheses involve bounds on a generalized Schwarzian derivative for harmonic mappings in terms 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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The authors were supported in part by FONDECYT Grant # 1110321.
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Chuaqui, M., Duren, P. & Osgood, B. Quasiconformal extensions to space of Weierstrass-Enneper lifts. JAMA 135, 487–526 (2018). https://doi.org/10.1007/s11854-018-0045-8
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DOI: https://doi.org/10.1007/s11854-018-0045-8