Approximation of analytic and Teichmüller spaces
Three problems related to the Teichmüller space theory are solved in this paper. The solution of these problems uses an interpolation by univalent functions with quasiconformal extension and an approximation of holomorphic functions by some automorphic (also holomorphic) forms with respect to certain Fuchsian groups. The interpolation theorem states that for any two given collections of different points \((z_ 1,...,z_ n)\) and \((w_ 1,...,w_ n)\) where the first contains the points lying on the unit circle counterclockwise and the second contains arbitrary points from the Riemann sphere \({\hat {\mathbb{C}}}\), there exist an univalent analytic function f on the closure of the unit disc such that \(f(z_ j)=w_ j\), \(j=1,...,n\). This result allows some generalizations. The proof employs a known interpolation result of V. Savin and of Clunie- Hallenbeck-Mac Gregor. This theorem is used to answer two questions posed in the book [\textit{S. Kruschkal} and \textit{R. Kühnau}: Quasikonforme Abbildungen - neue Methoden und Anwendungen (1985; Zbl 0539.30001)]. These questions concern with the structure of the spaces of quasiconformal deformations of an analytic function with quasiconformal extension. The obtained result shows that these spaces are biholomorphically equivalent to the Torelli spaces for punctured spheres. Using the above mentioned approximation of holomorphic functions, it is given a solution for the known problem on the inner radius of the Teichmüller space for the finitely generated Fuchsian groups of the first kind. This problem was posed elsewhere in the book: A crash course on the Kleinnian groups, \textit{L. Bers} (ed.) and \textit{I. Kra} (ed.), Lect. Notes Math. 400 (1974; Zbl 0279.00010). An independent solution of the last problem (and in a little more stronger form) is obtained recently by T. Nakanishi. Some open questions are posed and discussed also.
scientific article; zbMATH DE number 4175336