Apollonian isometries of planar domains are Möbius mappings

  • Peter Hästö
  • Zair Ibragimov


The Apollonian metric is a generalization of the hyperbolic metric, defined in a much larger class of open sets. Beardon introduced the metric in 1998, and asked whether its isometries are just the Möbius mappings. In this article we show that this is the case in all open subsets of the plane with at least three boundary points.

Math Subject Classifications

30F45 30C65 

Key Words and Phrases

Appolonian metric isometries 


  1. [1]
    Barbilian, D. Einordnung von Lobatschewsky’s Maßbestimmung in gewisse allgemeine Metrik der Jordanschen Bereiche,Casopsis Mathematiky a Fysiky 64, 182–183, (1934/35).Google Scholar
  2. [2]
    Barbilian, D. Asupra unui principiu de metrizare,Stud. Cercet. Mat. 10, 68–116, (1959).Google Scholar
  3. [3]
    Beardon, A.Geometry of Discrete Groups, Graduate text in mathematics 91, Springer-Verlag, New York, (1995).Google Scholar
  4. [4]
    Beardon, A. The Apollonian metric of a domain in ℝn, inQuasiconformal Mappings and Analysis, Duren, P., Heinonen, J., Osgood, B., and Palka, B., Eds., 91–108, Springer-Verlag, New York, (1998).Google Scholar
  5. [5]
    Berger, M.Geometry I, II, Springer-Verlag, Berlin, (1994).Google Scholar
  6. [6]
    Boskoff, W.-G.Hyperbolic Geometry and Barbilian Spaces, Istituto per la Ricerca di Base, Series of Monographs in Advanced Mathematics, Hardronic Press, Palm Harbor, FL, (1996).Google Scholar
  7. [7]
    Boskoff, W.-G.Varietăţi cu Structură Metrică Barbilian, (Romanian), Manifolds with Barbilian metric structure, Colecţia Biblioteca de Matematică Mathematics Library Collection, Ex Ponto, Editura, Constanţa, (2002).Google Scholar
  8. [8]
    Boskoff, W.-G. and Suceavă, B. Barbilian spaces: the history of a geometric idea, submitted (2005).Google Scholar
  9. [9]
    Chakerian, G. and Groemer, H. Convex bodies of constant width, inConvexity and its Applications, Gruber, P. and Wills, J., Eds., 49–96, Birkhäuser, Basel, (1983).Google Scholar
  10. [10]
    Ferrand, J. A characterization of quasiconformal mappings by the behavior of a function of three points, inProceedings of the 13th Rolf Nevalinna Colloquium, Joensuu, 1987; Laine, I., Rickman, S., and Sorvali, T., Eds., 110–123, Lecture Notes in Mathematics,1351, Springer-Verlag, New York, (1988).CrossRefGoogle Scholar
  11. [11]
    Gehring, F. and Hag, K. The Apollonian metric and quasiconformal mappings, inIn the Tradition of Ahlfors and Bers, Stony Brook, NY, 1998; Kra, I. and Maskit, B., Eds., 143–163, Contemp. Math. 256, Amer. Math. Soc., Providence, RI, (2000).Google Scholar
  12. [12]
    Gehring, F. and Palka, B. Quasiconformally homogeneous domains,J. Anal. Math. 30, 172–199, (1976).MathSciNetMATHGoogle Scholar
  13. [13]
    Hästö, P. The Apollonian metric: uniformity and quasiconvexity,Ann. Acad. Sci. Fenn. Math. 28, 385–414, (2003).MathSciNetMATHGoogle Scholar
  14. [14]
    Hästö, P. The Apollonian metric: limits of the approximation and bilipschitz properties,Abstr. Appl. Anal. 20, 1141–1158, (2003).CrossRefGoogle Scholar
  15. [15]
    Hästö, P. The Apollonian inner metric,Comm. Anal. Geom. 12(4), 927–947, (2004).MathSciNetMATHGoogle Scholar
  16. [16]
    Hästö, P. A new weighted metric: the relative metric II,J. Math. Anal. Appl. 301(2), 336–353, (2005).CrossRefMathSciNetMATHGoogle Scholar
  17. [17]
    Hästö, P. The Apollonian metric: the comparison property, bilipchitz mappings and thick sets, submitted (2005).Google Scholar
  18. [18]
    Hästö, P. and Ibragimov, Z. Apollonian isometries of regular domains are Möbius mappings, submitted (2005).Google Scholar
  19. [19]
    Herron, D., Ma, W., and Minda, D. A Möbius invariant metric for regions on the Riemann sphere, inFuture Trends in Geometric Function Theory, RNC Workshop, Jyväskylä 2003; Herron, D., Ed., 101–118, Rep. Univ. Jyväskylä Dept. Math. Stat.92, (2003).Google Scholar
  20. [20]
    Ibragimov, Z. The Apollonian metric, sets of constant width and Möbius modulus of ring domains, PhD Thesis, University of Michigan, Ann Arbor, MI, (2002).Google Scholar
  21. [21]
    Ibragimov, Z. On the Apollonian metric of domains in\(\overline {\mathbb{R}^n } \),Complex Var. Theory Appl. 48, 837–855, (2003).MathSciNetMATHGoogle Scholar
  22. [22]
    Ibragimov, Z. Conformality of the Apollonian metric,Comput. Methods Funct. Theory 3, 397–411, (2003).MathSciNetMATHGoogle Scholar
  23. [23]
    Kelly, P. Barbilian geometry and the Poincaré model,Amer. Math. Monthly 61, 311–319, (1954).CrossRefMathSciNetMATHGoogle Scholar
  24. [24]
    Kulkarni, R. and Pinkall, U. A canonical metric for Möbius structures and its applications,Math. Z. 216, 89–129, (1994).CrossRefMathSciNetMATHGoogle Scholar
  25. [25]
    Martin, G. and Osgood, B. The quasihyperbolic metric and associated estimates on the hyperbolic metric,J. Anal. Math. 47, 37–53, (1986).MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Rhodes, A. An upper bound for the hyperbolic metric of a convex domain,Bull. London Math. Soc. 29, 592–594, (1997).CrossRefMathSciNetMATHGoogle Scholar
  27. [27]
    Seittenranta, P. Möbius-invariant metrics,Math. Proc. Cambridge Philos. Soc. 125, 511–533, (1999).CrossRefMathSciNetMATHGoogle Scholar
  28. [28]
    Souza, P. Barbilian metric spaces and the hyperbolic plane (Spanish),Miscelánea Mat. 29, 25–42, (1999).MathSciNetMATHGoogle Scholar
  29. [29]
    Väisälä, J., Vuorinen, M., and Wallin, H. Thick sets and quasisymmetric maps,Nagoya Math. J. 135, 121–148, (1994).MathSciNetMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematics SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of Mathematical SciencesUniversity of CincinnatiCincinnati

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