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Journal d'Analyse Mathématique

, Volume 135, Issue 2, pp 487–526 | Cite as

Quasiconformal extensions to space of Weierstrass-Enneper lifts

  • M. Chuaqui
  • P. Duren
  • B. Osgood
Article
  • 17 Downloads

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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Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.P. Universidad Católica de ChileSantiagoChile
  2. 2.University of MichiganAnn ArborUSA
  3. 3.Stanford UniversityStanfordUSA

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