Abstract
In this paper we consider isometries of relative metrics. We characterize isometries of the j D metric and of Seittenranta’s metric, as well as of their generalizations. We also derive some inequalities and results on the geodesics of these metrics.
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References
A. Beardon, Geometry of Discrete Groups, Graduate Texts in Mathematics 91, Springer-Verlag, New York, 1995.
A. Beardon, The Apollonian metric of a domain in ℝn, in: P. Duren, J. Heinonen, B. Osgood and B. Palka (eds.), Quasiconformal Mappings and Analysis, Springer-Verlag, New York, 1998, 91–108.
L. Blumenthal, Distance Geometry. A Study of The Development of Abstract Metrics. With an Introduction by Karl Menger, Univ. of Missouri Studies Vol. 13, No. 2, Univ. of Missouri, Columbia, 1938.
D. Burago, Yu. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, Amer. Math. Soc., Providence, RI, 2001.
J. Ferrand, A characterization of quasiconformal mappings by the behavior of a function of three points, in: I. Laine, S. Rickman and T. Sorvali (eds.), Proceedings of the 13th Rolf Nevalinna Colloquium Joensuu, 1987; Lecture Notes in Mathematics Vol. 1351, Springer-Verlag, New York, 1988, 110–123.
F. Gehring and B. Osgood, Uniform domains and the quasihyperbolic metric, J. Anal. Math. 36 (1979), 50–74.
P. Hästö, A new weighted metric: the relative metric II, J. Math. Anal. Appl. 301 no. 2 (2005), 336–353.
P. Hästö, Gromov hyperbolicity of the jD and $⇔lde j_D$ metrics, Proc. Amer. Math. Soc. 134 (2006), 1137-1142..
P. Hästö and Z. Ibragimov, Apollonian isometries of planar domains are Möbius mappings, J. Geom. Anal. 15 no.2 (2005), 229–237.
—, Apollonian isometries of regular domains are Möbius mappings, submitted (2004).
P. Hästö and H. Lindén, Isometries of the half-apollonian metric, Complex Var. Theory Appl. 49 (2004), 405–415.
D. Herron, W. Ma and D. Minda, A Möbius invariant metric for regions on the Riemann sphere, in: D. Herron (ed.), Future Trends in Geometric Function Theory RNC Workshop, Jyväskylä 2003, Rep. Univ. Jyväskylä Dept. Math. Stat. 92 (2003), 101–118.
R. Kulkarni and U. Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994), 89–129.
G. Martin and B. Osgood, The quasihyperbolic metric and associated estimates on the hyperbolic metric, J. Anal. Math. 47 (1986), 37–53.
P. Seittenranta, Möbius-invariant metrics, Math. Proc. Cambridge Philos. Soc. 125 (1999), 511–533.
M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math., Vol. 1319, Springer, Berlin, 1988.
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PH was supported by the Research Council of Norway, Project 160192/V30; HL was supported in part by the Finnish Academy of Science and Letters.
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Hästö, P., Ibragimov, Z. & Lindén, H. Isometries of Relative Metrics. Comput. Methods Funct. Theory 6, 15–28 (2006). https://doi.org/10.1007/BF03321114
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DOI: https://doi.org/10.1007/BF03321114