Monatshefte für Mathematik

, Volume 117, Issue 1–2, pp 63–94 | Cite as

John disks and extension of maps

  • M. Ghamsari
  • R. Näkki
  • J. Väisalä


We show that a quasisymmetric map between the boundaries of two John disks can be extended to a quasiconformal map of the extended plane. Additional results on John disks are also given.

1991 Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ahlfors, L. V.: Lectures on Quasiconformal Mappings. Toronto-New York-London: Van Nostrand. 1966.Google Scholar
  2. [2]
    Aseyev, V. V.: Quasiconformal extension of quasi-Möbius mappings in the plane. Soviet Math. Dokl.38, 288–290 (1989).Google Scholar
  3. [3]
    Fernández, J. L., Heinonen, J., Martio, O.: Quasilines and conformal mappings. J. Analyse Math.52: 117–132 (1989).Google Scholar
  4. [4]
    Gehring, F. W.: Injectivity of local quasi-isometries. Comment. Math. Helv.57, 202–220 (1982).Google Scholar
  5. [5]
    Gehring, F. W.: Extension of quasiisometric embeddings of Jordan curves. Complex Variables Theory Appl.5, 245–263 (1986).Google Scholar
  6. [6]
    Gehring, F. W., Palka, B.: Quasiconformally homogeneous domains. J. Analyse Math.30, 172–199 (1976).Google Scholar
  7. [7]
    Heinonen, J., Näkki, R.: Quasiconformal distortion on arcs. To appear in J. Analyse Math.Google Scholar
  8. [8]
    John, F.: Rotation and strain. Comm. Pure Appl. Math.14, 391–413 (1961).Google Scholar
  9. [9]
    Kuratowski, K.: Topology. Vol. 2. New York: Academic Press. 1968.Google Scholar
  10. [10]
    Martio, O., Sarvas, J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math.4, 383–401 (1978).Google Scholar
  11. [11]
    Näkki, R.: Continuous boundary extension of quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math.511, 1–10 (1972).Google Scholar
  12. [12]
    Näkki, R., Väisälä, J.: John disks. Exposition. Math.9, 3–43 (1991).Google Scholar
  13. [13]
    Newman, M. H. A.: Elements of the Topology of Plane Sets of Points. Cambridge: University Press. 1961.Google Scholar
  14. [14]
    Pommerenke, C.: Lineare-invariante Familien analytischer Funktionen I. Math. Ann.155, 108–154 (1964).Google Scholar
  15. [15]
    Pommerenke, C.: Univalent Functions. Göttingen: Vandenhoeck and Ruprecht. 1975.Google Scholar
  16. [16]
    Rickman, S.: Characterization of quasiconformal arcs. Ann. Acad. Sci. Fenn. Ser. A I Math.395, 1–30 (1966).Google Scholar
  17. [17]
    Rickman, S.: Quasiconformally equivalent curves. Duke Math. J.36, 387–400 (1969).Google Scholar
  18. [18]
    Tukia, P., Väisälä, J.: Bilipschitz extensions of maps having quasiconformal extensions. Math. Ann.269, 561–572 (1984).Google Scholar
  19. [19]
    Väisälä, J.: Lectures onn-dimensional Quasiconformal Mappings. (Lecture Notes Math. 229). Berlin: Springer. 1971.Google Scholar
  20. [20]
    Väisälä, J.: Quasi-symmetric embeddings in euclidean spaces. Trans. Amer. Math. Soc.264, 191–204 (1981).Google Scholar
  21. [21]
    Väisälä, J.: Quasimöbius maps. J. Analyse Math.44, 218–234 (1984/85).Google Scholar
  22. [22]
    Väisälä, J.: Uniform domains. Tôhoku Math. J.40, 101–118 (1988).Google Scholar
  23. [23]
    Väisälä, J.: Quasiconformal maps of cylindrical domains. Acta Math.162, 201–225 (1989).Google Scholar
  24. [24]
    Väisälä, J.: Quasisymmetry and unions. Manuscripta Math.68, 101–111 (1990).Google Scholar
  25. [25]
    Whyburn, G. T.: Analytic Topology.Providence, R. I.: Amer. Math. Soc. 1942.Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • M. Ghamsari
    • 1
  • R. Näkki
    • 2
  • J. Väisalä
    • 3
  1. 1.Department of MathematicsChristopher Newport UniversityNewport NewsUSA
  2. 2.Jyväskylän yliopistoMatematiikan laitosJyväskyläFinland
  3. 3.Helsingin yliopistoMatematiikan laitosHelsinkiFinland

Personalised recommendations