Advertisement

Hyperbolic Type Metrics

  • Vesna Todorčević
Chapter
  • 177 Downloads

Abstract

The natural setup for our work here is a metric space (G, mG) where G is a subdomain of \(\mathbb {R}^n\,, n\ge 2\). For our studies, the distance mG(x, y), x, y ∈ G is required to take into account both how close the points x, y are to each other and the position of the points relative to the boundary ∂G. Metrics of this type are called hyperbolic type metrics and they are substitutes for the hyperbolic metric in dimensions n ≥ 3. The quasihyperbolic metric and the distance ratio metric are both examples of hyperbolic type metrics. A key problem is to study a quasiconformal mapping between metric spaces
$$\displaystyle f: (G, m_G) \to (f(G), m_{f(G)}) $$
and to estimate its modulus of continuity. We expect Holder continuity, but a concrete form of these results may differ from metric to metric. Another question is the comparison of the metrics to each other.

References

  1. 2.
    S.B. Agard, F.W. Gehring, Angles and quasiconformal mappings. Proc. Lond. Math. Soc. s3-14a, 1–21 (1965)MathSciNetCrossRefGoogle Scholar
  2. 4.
    L. Ahlfors, Möbius Transformations in Several Dimensions. Contemporary Mathematics: Introductory Courses (Mir, Moscow, 1986), 112 pp.Google Scholar
  3. 9.
    G.D. Anderson, M.K. Vamanamurty, M. Vuorinen, Sharp distortion theorems for quasiconformal mappings. Trans. Am. Math. Soc. 305(1), 95–111 (1988)MathSciNetCrossRefGoogle Scholar
  4. 10.
    G.D. Anderson, M.K. Vamanamurthy, M.K. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps. Canadian Mathematical Society Series of Monographs and Advanced Texts (Wiley, New York, 1997)zbMATHGoogle Scholar
  5. 24.
    A.F. Beardon, The Geometry of Discrete Groups. Graduate Texts in Mathematics, vol. 91 (Springer, New York, 1995)Google Scholar
  6. 25.
    A.F. Beardon, The Apollonian metric of a domain in \({\mathbb {R}^n}\), in Quasiconformal Mappings and Analysis, ed. by P. Duren, J. Heinonen, B. Osgood, B. Palka (Springer, New York, 1998), pp. 91–108Google Scholar
  7. 27.
    B. Bonfert-Taylor, R. Canary, E.C. Taylor, Quasiconformal homogeneity after Gehring and Palka. Comput. Methods Funct. Theory 14(2), 417–430 (2014)MathSciNetCrossRefGoogle Scholar
  8. 52.
    F.W. Gehring, K. Hag, The Ubiquitous Quasidisk. Mathematical Surveys and Monographs, vol. 184 (AMS, Providence, 2010)Google Scholar
  9. 53.
    F.W. Gehring, B.G. Osgood, Uniform domains and the quasihyperbolic metric. J. Anal. Math. 36, 50–74 (1979)MathSciNetCrossRefGoogle Scholar
  10. 54.
    F.W. Gehring, B.P. Palka, Quasiconformally homogeneous domains. J. Anal. Math. 30, 172–199 (1976)MathSciNetCrossRefGoogle Scholar
  11. 58.
    P. Hariri, M. Vuorinen, X. Zhang, Inequalities and bi-Lipschitz conditions for the triangular ratio metric. Rocky Mountain J. Math. 47(4), 1121–1148 (2017)MathSciNetCrossRefGoogle Scholar
  12. 59.
    P. Hästö, A new weighted metric: the relative metric. I. J. Math. Anal. Appl. 274, 38–58 (2002)MathSciNetCrossRefGoogle Scholar
  13. 61.
    P. Hästö, Z. Ibragimov, D. Minda, S. Ponnusamy, S.K. Sahoo, Isometries of some hyperbolic-type path metrics and the hyperbolic medial axis. In the tradition of Ahlfors-Bers, IV. Contemp. Math. 432, 63–74 (2007)Google Scholar
  14. 83.
    R. Klén, M. Vuorinen, X. Zhang, Quasihyperbolic metric and Möbius transformations. Proc. Am. Math. Soc. 142(1), 311–322 (2014)CrossRefGoogle Scholar
  15. 84.
    R. Klén, V. Todorčević, M. Vuorinen, Teichmüller’s problem in space. J. Math. Anal. Appl. 455(2), 1297–1316 (2017)MathSciNetCrossRefGoogle Scholar
  16. 86.
    V. Kojić, Metric spaces and quasiconformal mappings. Master Thesis, Belgrade (2007)Google Scholar
  17. 97.
    H. Lindén, Quasihyperbolic geodesics and uniformity in elementary domains. Dissertation, University of Helsinki, Helsinki, Annales Academiae Scientiarum Fennicae Mathematica Dissertationes No. 146 (2005), 50 pp.Google Scholar
  18. 98.
    V. Manojlović, Moduli of continuity of quasiregular mappings. Ph.D. Thesis, Belgrade (2008)Google Scholar
  19. 101.
    V. Manojlović, On conformally invariant extremal problems. Appl. Anal. Discrete Math. 3(1), 97–119 (2009)MathSciNetCrossRefGoogle Scholar
  20. 104.
    V. Manojlović, V. Vuorinen, On quasiconformal maps with identity boundary values. Trans. Am. Math. Soc. 363(5), 2467–2479 (2011)MathSciNetCrossRefGoogle Scholar
  21. 109.
    G.J. Martin, B.G. Osgood, The quasihyperbolic metric and the associated estimates on the hyperbolic metric. J. Anal. Math. 47, 37–53 (1986)MathSciNetCrossRefGoogle Scholar
  22. 112.
    O. Martio, J. Sarvas, Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Math. Ser. A I 4(2), 383–401 (1979)MathSciNetCrossRefGoogle Scholar
  23. 128.
    S. Ponnusamy, T. Sugawa, M.K. Vuorinen, Proceedings of International Workshop on Quasiconformal Mappings and their Applications, December 27, 2005–Jan 1, 2006, IIT Madras (2007)Google Scholar
  24. 129.
    T. Rado, P.V. Reichelderfer, Continuous transformations in analysis. With an introduction to algebraic topology. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. LXXV (Springer, Berlin, 1955), vii + 442 pp.Google Scholar
  25. 141.
    P. Seittenranta, Möbius-invariant metrics. Math. Proc. Cambridge Philos. Soc. 125(3), 511–533 (1999)MathSciNetCrossRefGoogle Scholar
  26. 152.
    O. Teichmüller, A displacement theorem of quasiconformal mapping, in Handbook of Teichmüller Theory. Translated from the German by Manfred Karbe. IRMA Lectures in Mathematics and Theoretical Physics, vol. VI (European Mathematical Society, Zürich, 2016), p. 27Google Scholar
  27. 158.
    M. Vuorinen, Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics, vol. 1319 (Springer, Berlin, 1988)CrossRefGoogle Scholar
  28. 159.
    M. Vuorinen, Quadruples and spatial quasiconformal mappings. Math. Z. 205(4), 617–628 (1990)MathSciNetCrossRefGoogle Scholar
  29. 160.
    M. Vuorinen, X. Zhang, Distortion of quasiconformal mappings with identity boundary values. J. Lond. Math. Soc. 90(3), 637–653 (2014)MathSciNetCrossRefGoogle Scholar
  30. 163.
    X. Zhang, Hyperbolic type metrics and distortion of quasiconformal mappings, Ph.D. Thesis, University of Turku 2013Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Vesna Todorčević
    • 1
    • 2
  1. 1.Faculty of Organizational SciencesUniversity of BelgradeBelgradeSerbia
  2. 2.Mathematical InstituteSerbian Academy of Sciences and ArtsBelgradeSerbia

Personalised recommendations