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. OsgoodEmail author


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Aharonov and U. Elias, Singular Sturm comparison theorems, J. Math. Anal. Appl. 371 (2010), 759–763.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    L. Ahlfors, Cross-ratios and Schwarzian derivatives in Rn, in Complex Analysis: Articles dedicated to Albert Pfluger on the Occasion of his 80th Birthday, Birkhäuser Verlag, Basel, 1989, pp. 1–15.Google Scholar
  3. [3]
    L. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975–978.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. Chuaqui, P. Duren, and B. Osgood, The Schwarzian derivative for harmonic mappings, J. Anal. Math. 91 (2003), 329–351.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    M. Chuaqui, P. Duren, and B. Osgood, Schwarzian derivative criteria for valence of analytic and harmonic mappings, Math. Proc. Camb. Phil. Soc. 143 (2007), 473–486.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Chuaqui, P. Duren, and B. Osgood, Univalence criteria for lifts of harmonic mappings to minimal surfaces, J. Geom. Anal. 17 (2007), 49–74.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Chuaqui and J. Gevirtz, Simple curves in Rn and Ahlfors’ Schwarzian derivative, Proc. Amer. Math. Soc. 132 (2004), 223–230.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Chuaqui and B. Osgood, Ahlfors-Weill extensions of conformal mappings and critical points of the Poincaré metric, Comment. Math. Helv. 69 (1994), 289–298.CrossRefzbMATHGoogle Scholar
  9. [9]
    M. Chuaqui, B. Osgood, and Ch. Pommerenke, John domains, quasidisks, and the Nehari class, J. Reine Angew. Math. 471 (1996), 77–114.MathSciNetzbMATHGoogle Scholar
  10. [10]
    P. Duren, Univalent Functions, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  11. [11]
    P. Duren, Harmonic Mappings in the Plane, Cambridge University Press, Cambridge, 2004.CrossRefzbMATHGoogle Scholar
  12. [12]
    C. Epstein, The hyperbolic Gauss map and quasiconformal reflections, J. Reine Angew. Math. 372 (1986), 96–135.MathSciNetzbMATHGoogle Scholar
  13. [13]
    F. W. Gehring and K. Hag, The Ubiquitous Quasidisk, Amer. Math. Soc., Providence, RI, 2012.CrossRefzbMATHGoogle Scholar
  14. [14]
    F. W. Gehring and Ch. Pommerenke, On the Nehari univalence criterion and quasicircles, Comment. Math. Helv. 59 (1984), 226–242.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    D. Minda, Euclidean circles of curvature for geodesics of conformal metrics, in Computational Methods and Function Theory 1997, World Sci. Publ., River Edge, NJ, 1999, pp. 397–403.CrossRefGoogle Scholar
  16. [16]
    Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    B. Osgood, Some properties of f′′/f′ and the Poincaré metric, Indiana Univ. Math. J. 31 (1982), 449–61.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    W. Sierra, John surfaces associated with a class of harmonic mappings, Ann. Acad. Sci. Fenn. Math. 38 (2013), 595–616.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D. Stowe, An Ahlfors derivative for conformal immersions, J. Geom. Anal. 25 (2013), 592–615.MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations