Advertisement

Journal of Mathematical Sciences

, Volume 244, Issue 1, pp 104–111 | Cite as

On the boundary behavior of quasiconformal mappings

  • Vladimir A. ZorichEmail author
Article
  • 2 Downloads

Abstract

We discuss some open questions of the theory of quasiconformal mappings related to the field of studies of Professor G. D. Suvorov. The present work is dedicated to his memory.

Keywords

Quasiconformal mapping boundary behavior ideal boundary prime ends Carnot–Carathéodory metric 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. Carathéodory, “Über die Begrenzung einfach zusammenhängender Gebiete,” Math. Ann., 73, 323–370 (1913).MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. A. Lavrent’ev, Selected Works. Mathematics and Mechanics [in Russian], Nauka, Moscow, 1990.zbMATHGoogle Scholar
  3. 3.
    M. A. Lavrent’ev, “On the continuity of univalent functions in closed domains,” Dokl. Akad. Nauk SSSR, 4, No. 5, 207–210 (1936) [in Russian].Google Scholar
  4. 4.
    G. D. Suvorov, Families of Flat Topological Mappings [in Russian], Sibir. Divis. of the AS of the USSR, Novosibirsk, 1965.Google Scholar
  5. 5.
    G. D. Suvorov, The Generalized Principle of Length and Area in the Theory of Mappings [in Russian], Naukova Dumka, Kiev, 1985.Google Scholar
  6. 6.
    V. A. Zorich, “The Carathéodory class and a spatial analog of the Koebe theorem,” Theory of Mappings, Its Generalizations, and Applications [in Russian], Naukova Dumka, Kiev, 1982, pp. 92–101.Google Scholar
  7. 7.
    F. W. Gehring, “Rings and quasiconformal mappings in space,” Trans. Amer. Math. Soc., 103, 353–393 (1962).MathSciNetCrossRefGoogle Scholar
  8. 8.
    G. D. Mostow, “Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms,” Publ. IHES, 34, 53–104 (1968).MathSciNetCrossRefGoogle Scholar
  9. 9.
    F. W. Gehring and J. Väisälä, “The coefficients of quasiconformality of domains in space,” Acta Math., 114, 1–70 (1965).MathSciNetCrossRefGoogle Scholar
  10. 10.
    V. A. Zorich, “Quasiconformal Immersions of Riemannian Manifolds and a Picard Type Theorem”, Funct. Anal. Appl., 34, No. 3, 188–196 (2000).MathSciNetCrossRefGoogle Scholar
  11. 11.
    C. Carathéodory, “Untersuhungen über die Grundlagen der Thermodynamik,” Math. Annalen, 67, 355–386 (1909).MathSciNetCrossRefGoogle Scholar
  12. 12.
    I. Hololpainen and S. Rickman, “Quasiregular mappings, Heisenberg group, and Picard’s theorem,” in: Proceeings of the Fourth Finnish–Polish Summer School in Complex Analysis, Jyväskylä, Finland, 1992, edited by J. Lawrinowicz et al., Jyväskylä Univ., Jyväskylä, 1993, pp. 25–35.Google Scholar
  13. 13.
    V. A. Zorich, “Quasi-conformal maps and the asymptotic geometry of manifolds”, Russian Math. Surveys, 57, No. 3, 437–462 (2002).MathSciNetCrossRefGoogle Scholar
  14. 14.
    V. A. Zorich, “Boundary behaviour of automorphisms of a hyperbolic space”, Russian Math. Surveys, 72, No. 4, 645–670 (2017).MathSciNetCrossRefGoogle Scholar
  15. 15.
    V. A. Zorich, Mathematical Aspects of Classical Thermodynamics [in Russian], MCCME, Moscow, 2019.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussian Federation

Personalised recommendations