The universal Teichmüller space, , is an infinite dimensional complex-analytic manifold modelled on a nonseparable Banach space. can be thought of as the space of (three-point-normalized) quasi-symmetric homeomorphisms of the unit circle . A homeomorphism, , is quasi-symmetric if and only if it allows a quasiconformal (q.c.) extension , where is the unit disc – the boundary of being . 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 in which allow a quasiconformal and diffeomorphic extension, , such that this extension 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 . For every , it is known that this harmonic and quasiconformal extension, , is a uniquely determined diffeomorphism of on itself. Consequently, one may associate to every in the standard holomorphic “Hopf differential” on arising from this harmonic mapping . That Hopf differential is a holomorphic function on which is a “bounded quadratic differential” (with respect to the Bers-Nehari norm): namely, is finite. Denote this Banach space of bounded holomorphic quadratic differentials on the disc by . The authors explain the inverse relationship: they show how to find the quasi-symmetric homeomorphism whose associated Hopf differential is any given function . Thus one has a bijective correspondence, say , between the Banach space and the subset of the universal Teichmüller space.
Note: When restricted to the -invariant quadratic differentials, , ( any torsion-free co-compact Fuchsian group), the above correspondence maps onto the finite-dimensional Teichmüller space . The correspondence then coincides with that studied by M. Wolf [J. Differ. Geom. 29, No. 2, 449-479 (1989; Zbl 0673.58014)]. This mapping is shown in this paper to be a real analytic diffeomorphism onto , where is shown to be an open subset of . But it is not clear whether this work provides a real analytic model for the universal Teichmüller space because the crucial question: “Is ?” remains unsolved.