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Harmonic representation of the universal Teichmüller space – the Paprocki space. (English) Zbl 1159.30334

Summary: To each element of the universal Teichmüller space represented by normalized quasihomographies of the unit circle \(T\) one associates uniquely an harmonic automorphism of the unit disk \(\Delta\), defined as Poisson integral \(P[f]\) of a given \(f\) in question. Hence, the universal Teichmüller space can be represented by certain family of harmonic automorphisms of the unit disk, denoted by \({\mathbf P}\) and called the Paprocki space. Among \(C^\infty\) and real analytic representations of the universal Teichmüller space this new one leads directly to two classes of analytic functions defined in the unit disk \(\Delta\) and by \({\mathcal P}_\Delta^1\) and \({\mathcal P}_\Delta^2\) called the conjugate Paprocki spaces of analytic functions. Formally, \({\mathcal P}_\Delta^1=\partial{\mathcal P}_\Delta\) and \({\mathcal P}_\Delta^2=\partial\overline{{\mathcal P}_\Delta}\). In this paper we study some basic properties of mappings from \({\mathcal P}_\Delta\) and functions from \({\mathcal P}_\Delta^1\) and \({\mathcal P}_\Delta^2\) classes.

MSC:

30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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