拟双曲度量与John圆

Quasihyperbolic Metric and John Disks

设D是R^2中的Jordan域,本文证明了D是b-John圆当且仅当存在常数c≥1,对任意的x_1,x_2∈D,有k_D(x_1,x_2)≤cH_D(x_1,x_2),这里kD(x_1,x_2)表示D中x_1与x_2二点的拟双曲距离,H_D(x_1,x_2)=1/2log(1+(l(γ))/(d(x_1,■D)))(1+(l(γ))/(d(x_2,■D))),其中l(γ)为D中连结x_1与x_2二点的拟双曲测地线的欧几里德长度.

Let D be a Jordan domain in R^2. In this paper, we prove that D is a b-John disk if and only if there exists a constant c ≥ 1 such that kD（x1,x2） ≤ cHD（x1,x2） for all x1, x2 ∈D. Here kD（x1, x2） is the quasihyperbolic metric in D, HD（x1, x2） = 1/2log（1＋（l（γ））/（d（x1,δD）））（1＋（l（γ））/（d（x2,δD）））, where γ, is the quasi-hyperbolic geodesic which joins x1 and x2 in D and l（γ） is the Euclidean length of γ.