A commentary on Teichmüller’s paper <em>Veränderliche Riemannsche Flächen</em>

Abstract

This is a commentary on Teichmullers' paper ''Veranderliche Riemannsche Flachen (Variable Riemann Surfaces), published in 1944. This paper is the last one that Teichmuller wrote on the problem of moduli. At most places the paper contains ideas and no technical details. The author presents a completely new approach to Teichmuller space, compared to the approach he took in his first seminal paper ''Extremale quasikonforme Abbildungen und quadratische Differentiale and its sequel ''Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flachen in which he completed some of the the results stated in the former. In the paper ''Extremale quasikonforme ..., Teichmuller led the foundations of what we call today Teichmuller theory (but without the complex structure), defining its metric and introducing in that theory the techniques of quasiconformal mappings and of quadratic differentials as essential tools. In the present paper, the approach is more abstract, through complex analytic geometry. Teichmuller space, equipped with its complex-analytic structure, is characterized here by a certain universal property. Among the other ideas and results contained in the paper, we mention the following: (1) The existence and uniqueness of the universal Teichmuller curve, rediscovered later on by Ahlfors and by Bers. At the same time, this introduced the first fibre bundle over Teichmuller space. (2) The proof of the fact that the automorphisms group of the univeral Teichmuller curve is the extended mapping class group. (3) The idea of a fine moduli space. (4) The idea of using the period map to define a complex structure on Teichmuller space.