Douady-Earle延拓中用到的拟共形映照参数表示

Parametric Representation of Douady-Earle Quasiconformal Extension

假设f^(μ(z))(z)是单位圆D到自身保持-1,i,1不动,具有复特征μ(z)的Douady-Earle延拓。借助于上半平面到自身保持0,1,∞三点不动的拟共形映射的参数表示,利用单位圆到上半平面的共形映射,给出Douady-Earle延拓f^(μ(z))(z)的参数表示。

Suppose f^(μ(z))(z)is a Douady-Earle extension of the unit disk D onto itself with the complex dilatationμ(z),which kept-1,i,1 fixed.With the help of the parametric representation of the quasiconformal mapping of the upper half plane onto itself which kept 0,1,∞fixed,by using the conformal mapping from the unit disk to the upper half plane,the parametric representation of the Douady-Early extension f^(μ(z))(z)is given.