Apollonian度量与Apollonian边界条件

The Apollonian Metric and Apollonian Boundary Condition

证明了(1)■中真子域D上的Apollonian度量αD是拟共形映射的拟不变量;(2)■中严格一致域是拟共形不变的;(3)■中的Jordan域D是拟圆当且仅当D是严格一致域,作为应用,进一步得到了Apollonian边界条件,拟共形映射和局部Lipschitz映射之间的关系。

This paper proves： （1） The Apollonian metric aD of a proper subdomain D ∈ R^-n is a quasi-invariant metric under quasiconformal mappings; （2） The strictly uniform domains of R^-n are invariant under quasiconformal mappings; （3） A Jordan domain D ∈R^-2 is a quasidisk if and only if D is a strictly uniform domain. Furthermore, as applications, some relations between Apollonian boundary condition, quasiconformal mappings and lo- cally Lipschitz mappings is obtained.