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The Apollonian metric and quasiconformal mappings. (English) Zbl 0964.30024

Kra, Irwin (ed.) et al., In the tradition of Ahlfors and Bers. Proceedings of the first Ahlfors-Bers colloquium, State University of New York, Stony Brook, NY, USA, November 6-8, 1998. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 256, 143-163 (2000).
The Apollonian metric \(a_D\) is defined on any proper subdomain of the extended complex plane (or even in higher dimensions). It was introduced by Barbilian about 1935 but named Apollonian by Beardon in 1998. In the extended complex plane, it is defined in a Möbius-invariant way without intervention of conformal mapping by the formula \[ a_D(z_1, z_2)= \sup_{w_1,w_2\in \partial D} \log\Biggl( {|z_1- w_1||z_2- w_2|\over|z_1- w_2||z_2- w_1|}\Biggr), \] where \(z_1,z_2\in D\). It is particularly attractive because, although it is defined without intervention of conformal mapping, it is precisely the hyperbolic metric when \(D\) is convex and gives the following sharp lower bound for the hyperbolic metric \(h_D\) whenever \(D\) is simply connected of hyperbolic type: \[ a_D(z_1, z_2)\leq 2h_D(z_1, z_2). \] The authors investigate the circumstances under which the metric \(a_D\) also supplies an upper bound of the form \(h_D(z_1, z_2)\leq ca_D(z_1, z_2)\) and find that this is so for simply connected domains \(D\) of hyperbolic type if and only if \(D\) is a quasidisk. They also study the Apollonian isometries with domain of definition a quasidisk and find that the Apollonian is quasiconformal if and only if the target domain is a quasidisk. They have an analogous result about conformal mappings between quasidisks, and they characterize the domains of hyperbolic type all of whose points are connected by Apollonian geodesics as Möbius images of the standard disk.
For the entire collection see [Zbl 0941.00017].
Reviewer: J.W.Cannon (Provo)

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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