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.
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Communicated by Steven Krantz
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Hästö, P., Ibragimov, Z. Apollonian isometries of planar domains are Möbius mappings. J Geom Anal 15, 229–237 (2005). https://doi.org/10.1007/BF02922194
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DOI: https://doi.org/10.1007/BF02922194