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Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space. (English) Zbl 0873.32019
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, {\it L. Ahlfors}, “Lectures on quasiconformal mappings”, Van Nostrand (1966; Zbl 0138.06002), or, {\it 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 ${\cal {B}}$, 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 ${\cal {B}}$ then coincides with that studied by {\it M. Wolf} [J. Differ. Geom. 29, No. 2, 449-479 (1989; Zbl 0673.58014)]. This mapping ${\cal B}: B(D) \rightarrow {T'}$ 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.

32G15Moduli of Riemann surfaces, Teichmüller theory
30F60Teichmüller theory