关于Gr(o)zch问题的一个注记及万有Teichmüller空间一性质的证明

摘要

朱华成等在"Gr(o)zch问题的域内特征"一文中给出了拟共形映射的Schwarz型引理:设f(z)是单位圆上的k-拟共形自同胚,若f(0)=0,lim z→G (│f(z)│/│z│1/k)=1,则:f(z)=eiθ z│z│1/k-1,θ是实数.原文等价证明部分对θ是实数的证明未说明关键点h.(θ)跟r无关(其中z=reiθ),本文做了补充研究;另给出了文献[3]中定理2.1的简洁证明.%Zhu Hua cheng, Zhou Ze min and He Cheng qi have given the characterization of Grotzsch's problem:Suppose that f(z) is K -quasiconformal in the unit disk Δ.ff and f(0) =0,and lim |f(z)| = 1 thenf(z) =e iθzlz x→G |z| 1/k |(1/k) -1, θ ∈ R, (z = reiθ).But during the proof of problem, it is a non-complete proof that is real.In this paper, we give a complete the proof of it, and often the concise proof.For theorem 2.1 in Wang zhe' s paper“ The distance between different component of the universal Teichmüller space ”.