Quasiconformal isotopies

  • Clifford J. Earle
  • Curt McMullen
Conference paper
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 10)


Let X be a hyperbolic Riemann surface or orbifold, possibly of infinite topological complexity. Let Ø: X → X be a quasiconformal map. We show the following conditions are equivalent (§1):
  1. (a)

    Ø has a lift to the universal cover Δ which is the identity on S1;

  2. (b)

    Ø is homotopic to the identity rel the ideal boundary of X; and

  3. (c)

    Ø is isotopic to the identity rel ideal boundary, through uniformly quasiconformal maps.





Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ahlf]
    Ahlfors, L., “Conformal Invariants,” McGraw-Hill, 1973.MATHGoogle Scholar
  2. [AB]
    Ahlfors, L., and Bers, L., Riemann mapping theorem for variable metrics, Annals of Math 72 (1960), 385–404.CrossRefMATHMathSciNetGoogle Scholar
  3. [B1]
    Bers, L., Uniformization, moduli and Kelinian groups, Bull. London Math. Soc. 4 (1972), 257–300.CrossRefMATHMathSciNetGoogle Scholar
  4. [B2]
    Bers, L., The moduli of Kleinian groups, Russian Math Surveys 29 (1974), 88–102.CrossRefMATHMathSciNetGoogle Scholar
  5. [B3]
    Bers, L., On Sullivan’s proof of the finiteness theorem and the eventual periodicity theorem, Preprint.Google Scholar
  6. [BG]
    Bers, L., and Greenberg, L., Isomorphisms between Teichmüller spaces, in Advances in the Theory of Riemann Surfaces, Princeton: Annals of Math Studies 66 (1971), 53–79.MathSciNetGoogle Scholar
  7. [BR]
    Bers, L., and Royden, H.L., Holomorphic families of injections, Acta Math. 157 (1986), 259–286.CrossRefMATHMathSciNetGoogle Scholar
  8. [Bon]
    Bonahon, F., Bouts des varieties hyperbolique de dimension trois, To appear.Google Scholar
  9. [DE]
    Douady, A., and Earle, C., Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), 23–48.CrossRefMATHMathSciNetGoogle Scholar
  10. [DH]
    Douady, A., and Hubbard, J., On the dynamics of polynomial-like mappings, Ann. Sci. Ec. Norm. Sup. 18 (1985), 287–344.MATHMathSciNetGoogle Scholar
  11. [EEl]
    Earle, C., and Eells, J., On the differential geometry of Teichmüller spaces, J. Analyse Math. 19 (1967), 35–52.CrossRefMATHMathSciNetGoogle Scholar
  12. [EE2]
    Earle, C., and Eells, J., A fibre bundle description of Teichmüller theory, J. Diff. Geom.3 (1969), 19–43.Google Scholar
  13. [Eps]
    Epstein, D.B.A., Curves on 2-manifolds and isotopies, Acta Math. 115 (1966), 83–107.CrossRefMATHMathSciNetGoogle Scholar
  14. [FRW]
    Fitzgerald, C.H., Rodin, B., and Warschawski, S.E., Estimates for the harmonic measure of a continuum in the unit disk, Trans. AMS 287 (1985), 681–685.CrossRefMATHMathSciNetGoogle Scholar
  15. [Gar]
    Gardiner, F., A theorem of Bers and Greenberg for infinite dimensional Teichmüller spaces, These proceedings.Google Scholar
  16. [Mar]
    Marden, A., On homotopic mappings of Riemann surfaces, Annals of Math. 90 (1969), 1–8.CrossRefMATHMathSciNetGoogle Scholar
  17. [Sul]
    Sullivan, D., Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Annals of Math. 122 (1985), 401–418.CrossRefGoogle Scholar
  18. [Thur]
    Thurston, W., Geometry and Topology of Three Manifolds, Princeton lecture notes.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Clifford J. Earle
    • 1
    • 2
  • Curt McMullen
    • 2
  1. 1.Cornell UniversityIthacaUSA
  2. 2.M.S.R.I.BerkeleyUSA

Personalised recommendations