Quasiconformal isotopies

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

Abstract

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.

     

Keywords

Riemann Surface Harmonic Measure Kleinian Group Fuchsian Group Ideal Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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

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