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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 S 1 , is quasi-symmetric if and only if it allows a quasiconformal (q.c.) extension H:DD, 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 ' . For every hT ' , 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 φ on D which is a “bounded quadratic differential” (with respect to the Bers-Nehari norm): namely, sup D |(1-|z| 2 ) 2 φ(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 φB(D). Thus one has a bijective correspondence, say , 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)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 then coincides with that studied by M. Wolf [J. Differ. Geom. 29, No. 2, 449-479 (1989; Zbl 0673.58014)]. This mapping :B(D)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.


MSC:
32G15Moduli of Riemann surfaces, Teichmüller theory
30F60Teichmüller theory