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Apollonian isometries of planar domains are Möbius mappings. (English) Zbl 1081.30039

\(D\) is an open nonempty subset of the Riemann sphere \(\overline{\mathbb R}^2\), with \(D\not=\overline{\mathbb R}^2\). The Apollonian metric on \(D\) is defined by \[ \alpha_D(x,y)=\sup_{a,b\in\partial D} | a,y,x,b| \] where \(| a,y,x,b|\) denotes cross product. \(\alpha_D\) is in general a semi-metric, and it is a metric if and only if the complement of \(D\) is not contained in a circle. The Apollonian metric was discovered by D. Barbilian in his paper [“Einordnung von Lobatschewskys Massbestimmung in gewisse allgemeine Metrik der Jordanschen Bereiche”, Casopis Praha 64, 182–183 (1935; JFM 61.0601.02)], and later on rediscovered independently by A. Beardon (who coined the name “Apollonoan”) in his paper [“The Apollonian metric of a domain in \(\mathbb{R}^n\)”, P. Duren et al. (ed.), Quasiconformal mappings and analysis, Proceedings of an international symposium, Ann Arbor, 1995, Springer, 91–108 (1998; Zbl 0890.30030)]. In fact, the Apollonian metric is defined more generally for all proper domains in \(\overline{\mathbb R}^n\). It follows immediately from the definition that \(\alpha_D\) is invariant by Möbius transformations, and Beardon asked in the same paper whether every isometry of \(\alpha_D\) is a Möbius transformation. In the paper under review, the authors give an affirmative answer to Beardon’s question in the case of all open subsets of the plane which have at least three points on their boundary. The authors note, by giving an example, that the result is false if one removes the condition that the boundary of the domains contains at least three points.

MSC:

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
30C35 General theory of conformal mappings
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[1] Barbilian, D., Einordnung von Lobatschewsky’s Maßbestimmung in gewisse allgemeine Metrik der Jordanschen Bereiche, Casopsis Mathematiky a Fysiky, 64, 182-183 (1934) · JFM 61.0601.02
[2] Barbilian, D., Asupra unui principiu de metrizare, Stud. Cercet. Mat., 10, 68-116 (1959)
[3] Beardon, A., Geometry of Discrete Groups, Graduate text in mathematics 91 (1995), New York: Springer-Verlag, New York
[4] Beardon, A.; Duren, P.; Heinonen, J.; Osgood, B.; Palka, B., The Apollonian metric of a domain in ℝ^n, Quasiconformal Mappings and Analysis, 91-108 (1998), New York: Springer-Verlag, New York · Zbl 0890.30030
[5] Berger, M., Geometry I, II (1994), Berlin: Springer-Verlag, Berlin
[6] Boskoff, W.-G., Hyperbolic Geometry and Barbilian Spaces, Istituto per la Ricerca di Base, Series of Monographs in Advanced Mathematics (1996), Palm Harbor, FL: Hardronic Press, Palm Harbor, FL · Zbl 0851.51002
[7] Boskoff, W.-G., Varietăţi cu Structură Metrică Barbilian, (Romanian), Manifolds with Barbilian metric structure, Colecţia Biblioteca de Matematică Mathematics Library Collection (2002), Ex Ponto: Editura, Constanţa, Ex Ponto · Zbl 1015.51001
[8] Boskoff, W.-G. and Suceavă, B. Barbilian spaces: the history of a geometric idea, submitted (2005). · Zbl 1126.01015
[9] Chakerian, G.; Groemer, H.; Gruber, P.; Wills, J., Convex bodies of constant width, Convexity and its Applications, 49-96 (1983), Basel: Birkhäuser, Basel · Zbl 0518.52002
[10] Ferrand, J.; Laine, I.; Rickman, S.; Sorvali, T., A characterization of quasiconformal mappings by the behavior of a function of three points, Proceedings of the 13th Rolf Nevalinna Colloquium, Joensuu, 1987, 110-123 (1988), New York: Springer-Verlag, New York · Zbl 0661.30015 · doi:10.1007/BFb0081247
[11] Gehring, F.; Hag, K.; Kra, I.; Maskit, B., The Apollonian metric and quasiconformal mappings, In the Tradition of Ahlfors and Bers, Stony Brook, NY, 1998, 143-163 (2000), Providence, RI: Amer. Math. Soc., Providence, RI · Zbl 0964.30024
[12] Gehring, F.; Palka, B., Quasiconformally homogeneous domains, J. Anal. Math., 30, 172-199 (1976) · Zbl 0349.30019
[13] Hästö, P., The Apollonian metric: uniformity and quasiconvexity, Ann. Acad. Sci. Fenn. Math., 28, 385-414 (2003) · Zbl 1029.30027
[14] Hästö, P., The Apollonian metric: limits of the approximation and bilipschitz properties, Abstr. Appl. Anal., 20, 1141-1158 (2003) · Zbl 1067.30082 · doi:10.1155/S1085337503309042
[15] Hästö, P., The Apollonian inner metric, Comm. Anal. Geom., 12, 4, 927-947 (2004) · Zbl 1146.30025
[16] Hästö, P., A new weighted metric: the relative metric II, J. Math. Anal. Appl., 301, 2, 336-353 (2005) · Zbl 1069.54019 · doi:10.1016/j.jmaa.2004.07.034
[17] Hästö, P. The Apollonian metric: the comparison property, bilipchitz mappings and thick sets, submitted (2005).
[18] Hästö, P. and Ibragimov, Z. Apollonian isometries of regular domains are Möbius mappings, submitted (2005). · Zbl 1081.30039
[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). · Zbl 1058.30020
[20] Ibragimov, Z., The Apollonian metric, sets of constant width and Möbius modulus of ring domains, PhD Thesis (2002), Ann Arbor, MI: University of Michigan, Ann Arbor, MI
[21] Ibragimov, Z., On the Apollonian metric of domains in \(\overline{\mathbb{R}^n } \), Complex Var. Theory Appl., 48, 837-855 (2003) · Zbl 1043.30026
[22] Ibragimov, Z., Conformality of the Apollonian metric, Comput. Methods Funct. Theory, 3, 397-411 (2003) · Zbl 1055.30039
[23] Kelly, P., Barbilian geometry and the Poincaré model, Amer. Math. Monthly, 61, 311-319 (1954) · Zbl 0055.41002 · doi:10.2307/2307467
[24] Kulkarni, R.; Pinkall, U., A canonical metric for Möbius structures and its applications, Math. Z., 216, 89-129 (1994) · Zbl 0813.53022 · doi:10.1007/BF02572311
[25] Martin, G.; Osgood, B., The quasihyperbolic metric and associated estimates on the hyperbolic metric, J. Anal. Math., 47, 37-53 (1986) · Zbl 0621.30023 · doi:10.1007/BF02792531
[26] Rhodes, A., An upper bound for the hyperbolic metric of a convex domain, Bull. London Math. Soc., 29, 592-594 (1997) · Zbl 0914.30005 · doi:10.1112/S0024609397003391
[27] Seittenranta, P., Möbius-invariant metrics, Math. Proc. Cambridge Philos. Soc., 125, 511-533 (1999) · Zbl 0917.30015 · doi:10.1017/S0305004198002904
[28] Souza, P., Barbilian metric spaces and the hyperbolic plane (Spanish), Miscelánea Mat., 29, 25-42 (1999) · Zbl 1044.51013
[29] Väisälä, J.; Vuorinen, M.; Wallin, H., Thick sets and quasisymmetric maps, Nagoya Math. J., 135, 121-148 (1994) · Zbl 0803.30016
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