The Apollonian Metric: Limits of the Comparison and Bilipschitz Properties

The Apollonian metric is a generalization of the hyperbolic metric introduced by Beardon [2]. It is defined in arbitrary domains in Rn and is Möbius invariant. Another advantage over the well-known quasihyperbolic metric [8] is that it is simpler to evaluate. On the downside, points cannot generally be connected by geodesics of the Apollonian metric. This paper is the last in a series of four papers on the Apollonian metric, the first three being [9, 10, 11]. Other authors who have approached this metric from the same perspective, providing the incentive for this investigation, are Rhodes [13], Seittenranta [14], Gehring and Hag [5, 6], and Ibragimov [12]. As becoming of a concluding paper, we will return here to the beginning and take a new look at the comparison and bilipschitz properties considered in [10]. Using results from [9], we are able to answer a question posed to the author by M. Vuorinen, which led to the start of this investigation, namely: under what circumstances are Euclidean bilipschitz with small distortion also Apollonian bilipschitz mappings with small distortion? This question can be seen as a step towards answering the question asked in [2] by Beardon about the isometries of the Apollonian metric, since the comparison condition has previously been shown to imply quite some regularity of the Apollonian metric (cf., e.g., the proof of [10, Theorem 1.6]).