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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, $h:{S}^{1}\to {S}^{1}$, is quasi-symmetric if and only if it allows a quasiconformal (q.c.) extension $H:D\to D$, where $D$ is the unit disc – the boundary of $D$ being ${S}^{1}$. For this basic background, see, for instance, L. Ahlfors, “Lectures on quasiconformal mappings”, Van Nostrand (1966; Zbl 0138.06002), or, 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 ${T}^{\text{'}}$. For every $h\in {T}^{\text{'}}$, 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 ${T}^{\text{'}}$ 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, $su{p}_{D}|\left(1-{|z|}^{2}{\right)}^{2}\phi \left(z\right)|$ is finite. Denote this Banach space of bounded holomorphic quadratic differentials on the disc by $B\left(D\right)$. The authors explain the inverse relationship: they show how to find the quasi-symmetric homeomorphism $h$ whose associated Hopf differential is any given function $\phi \in B\left(D\right)$. Thus one has a bijective correspondence, say $ℬ$, between the Banach space $B\left(D\right)$ and the subset ${T}^{\text{'}}$ of the universal Teichmüller space.

Note: When restricted to the $G$-invariant quadratic differentials, ${B}_{G}\left(D\right)\subset B\left(D\right)$, ($G$ any torsion-free co-compact Fuchsian group), the above correspondence maps ${B}_{G}\left(D\right)$ onto the finite-dimensional Teichmüller space $T\left(G\right)$. The correspondence $ℬ$ then coincides with that studied by M. Wolf [J. Differ. Geom. 29, No. 2, 449-479 (1989; Zbl 0673.58014)]. This mapping $ℬ:B\left(D\right)\to {T}^{\text{'}}$ is shown in this paper to be a real analytic diffeomorphism onto ${T}^{\text{'}}$, where ${T}^{\text{'}}$ 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 ${T}^{\text{'}}=T$?” remains unsolved.

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory 30F60 Teichmüller theory