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Apollonian isometries of planar domains are Möbius mappings

  • Peter Hästö
  • Zair Ibragimov
Article

Abstract

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 

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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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