A commentary on Teichmüller’s paper <i>Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen</i>

Abstract

This is a mathematical commentary on Teichmüller's paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces) [20], (1943). This paper is among the last (and may be the last one) that Teichmüller wrote on the theory of moduli. It contains the proof of the so-called Teichmüller existence theorem for a closed surface of genus g ≥ 2. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichmüller space) and a space of equivalence classes of certain Fuchsian groups (the so-called Fricke space). 1 After that, he defines a map between the latter and the Euclidean space of dimension 6g − 6. Using Brouwer's theorem of invariance of domain, he shows that this map is a homeomorphism. This involves in particular a careful definition of the topologies of Fricke space, the computation of its dimension, and comparison results between hyperbolic distance and quasiconformal dilatation. The use of the invariance of domain theorem is in the spirit of Poincaré and Klein's use of the so-called "continuity principle" in their attempts to prove the uniformization theorem.

A commentary on Teichmüller’s paper <i>Extremale quasikonforme Abbildungen und quadratische Differentiale</i>

“…Item (1) explains how to define, for a given Riemannian metric, the associated dilatation quotient. Item (2) implies that if there exists a map homotopic to the identity such that the dilatation quotient of its associated metric is constant, then the map is extremal. Item (3) says that there always exists an extremal map.…”

“…For a more global view of Teichmüller's contribution on moduli of Riemann surfaces, the reader is also referred to his other papers [39], [37], [41], [32], [34], [40] and the corresponding commentaries [14], [5], [11], [2], [10], [3], as well as the survey papers [1], [19], [4].…”

On Grothendieck’s construction of Teichmüller space

“…12 Teichmüller was aware of the equivalence of the various notions of markings. In his paper [45] (see also the commentary [1]), he uses three sorts of markings: homotopy classes of homeomorphisms, isotopy classes of homeomorphisms, and the choice of a basis of the fundamental group. These equivalences are deep theorems in the topology of surfaces, and Teichmüller attributes them to Mangler [36].…”

A Commentary on Teichmüller’s paper <i>Vollständige Lösung einer Extremalaufgabe der quasikonformen Abbildung</i>

“…In this case, one of the vertices (the one with the re-entering angle) is not considered as a distinguished point, and the hexagon becomes the conformal image of a pentagon. 1 In [9], Teichmüller promises to give later on a proof in the most general case of surfaces of finite type (orientable or not, with or without boundary, with or without distinguished points in the interior and/or on the boundary). His project was not realized since he died soon later.…”

“…Let P be a pentagon determined by a pair (p 2 , p 4 ) ∈ (0, 1) × (1, ∞). Using the formula given by (1), we obtain, from ϕ (a real number modulo 2π), a new coordinate ζ in which the pentagon has one of the desired forms, that is, either a hexagon in the plane whose distinguished points are the five salient vertices, or a rectangular hexagon. In either case, we obtain an equivalence class of such hexagons called S. Teichmüller defines, for K ≥ 1, a map (which is called now a Teichmüller map), between two Euclidean hexagons which in natural local coordinates has the form…”