圆上的Apollonian度量与双曲度量

Apollonian metric and hyperbolic metric on disk

设D是R2中至少包含三个边界点的单连通区域,对任意x,y∈D,αD(x,y)和hD(x,y)分别表示D中关于x,y两点的Apollonian度量和双曲度量.文中肯定并证明了A.F.Beardon于1998年提出的猜想:对任意x,y∈D,αD(x,y)=hD(x,y)成立的充要条件是D为圆.

If D ~2 is a simply connected domain with boundary containing at least three points,then for any x,y∈D,α_D(x,y) and h\-D(x,y) represent the Apollonian metric and hyperbolic metric in D with respect to x and y respectively.In this paper, the conjecture which was given by Beardon in 1998 is affirmed and proved:for any x,y∈D,α_D(x,y)=h\-D(x,y) if and only if D be a disk.