The distance between different components of the universal Teichmüller space
In this paper the author considers a class of maps. The model of the universal Teichmüller space considered by the author in this paper is the set of conformal maps \(f\) defined on the disc with \(f(0)=0\), \(f'(0)=1\) and possessing a quasi-conformal extension to the Riemann sphere. One considers the norm \(\sup (1-| z| ^{2}) | f''(z)/f'(z)| \) where the supremum is taken over the unit disc. It has been shown by \textit{I. V. Zhuravlev} [``Model of the universal Teichmüller space'', Sib. Math. J. 27, 691--697 (1986); translation from Sib. Mat. Zh. 27, No. 5 (159), 75--82 (1986; Zbl 0616.30023)] that in the induced topology the universal Teichmüller space is disconnected and one can describe the components. Moreover one can construct for each component a special element called here the \textit{centre}. The author shows here that the distance between any two components is zero and that the distance between the centres of two distinct components is 2.
scientific article; zbMATH DE number 5008960