Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space
The universal Teichmüller space, \(T\), is an infinite dimensional complex-analytic manifold modelled on a nonseparable Banach space. \(T\) can be thought of as the space of (three-point-normalized) quasi-symmetric homeomorphisms of the unit circle \(S^1\). A homeomorphism, \(h: S^1 \rightarrow S^1\), is quasi-symmetric if and only if it allows a quasiconformal (q.c.) extension \(H: D \rightarrow D\), where \(D\) is the unit disc -- the boundary of \(D\) being \(S^1\). For this basic background, see, for instance, \textit{L. Ahlfors}, ``Lectures on quasiconformal mappings'', Van Nostrand (1966; Zbl 0138.06002), or, \textit{S. Nag} ``The Complex analytic theory of Teichmüller spaces'', Wiley-Interscience (1988; Zbl 0667.30040). Consider the set of those \(h\) in \(T\) which allow a quasiconformal and diffeomorphic extension, \(H\), such that this extension \(H\) is furthermore required to be a harmonic self-mapping of the unit disc (the disc being equipped with the Poincaré hyperbolic metric). Denote this subset of the universal Teichmüller space by \(T'\). For every \(h \in T'\), it is known that this harmonic and quasiconformal extension, \(H = H_{h}\), is a uniquely determined diffeomorphism of \(D\) on itself. Consequently, one may associate to every \(h\) in \(T'\) the standard holomorphic ``Hopf differential'' on \(D\) arising from this harmonic mapping \(H_{h}\). That Hopf differential is a holomorphic function \(\phi\) on \(D\) which is a ``bounded quadratic differential'' (with respect to the Bers-Nehari norm): namely, \(sup_{D}|(1-|z|^{2})^{2}{\phi}(z) |\) is finite. Denote this Banach space of bounded holomorphic quadratic differentials on the disc by \(B(D)\). The authors explain the inverse relationship: they show how to find the quasi-symmetric homeomorphism \(h\) whose associated Hopf differential is any given function \(\phi \in B(D)\). Thus one has a bijective correspondence, say \({\mathcal {B}}\), between the Banach space \(B(D)\) and the subset \(T'\) of the universal Teichmüller space. Note: When restricted to the \(G\)-invariant quadratic differentials, \(B_{G}(D) \subset B(D)\), (\(G\) any torsion-free co-compact Fuchsian group), the above correspondence maps \(B_{G}(D)\) onto the finite-dimensional Teichmüller space \(T(G)\). The correspondence \({\mathcal {B}}\) then coincides with that studied by \textit{M. Wolf} [J. Differ. Geom. 29, No. 2, 449-479 (1989; Zbl 0673.58014)]. This mapping \({\mathcal B}: B(D) \rightarrow {T'}\) is shown in this paper to be a real analytic diffeomorphism onto \(T'\), where \(T'\) is shown to be an open subset of \(T\). But it is not clear whether this work provides a real analytic model for the universal Teichmüller space because the crucial question: ``Is \(T'=T\)?'' remains unsolved.
scientific article; zbMATH DE number 894386
scientific article