A Commentary on Teichmüller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen"

Annette A'Campo-Neuen (1) , Norbert A'Campo (1) , Vincent Alberge (2, 3) , Athanase Papadopoulos (2, 3)

1 Unibas - Université de Bâle = University of Basel = Basel Universität
2 IRMA - Institut de Recherche Mathématique Avancée
3 The Graduate Center - CUNY Graduate Center

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). 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 at least 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). 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.

Mots clés: Fuchsian group.; Teichmüller existence theorem; extremal quasiconformal mapping; Teichmüller space; Fricke space

Domaines: Topologie géométrique [math.GT]; Variables complexes [math.CV]

Annette A'Campo-Neuen, Norbert A'Campo, Vincent Alberge, Athanase Papadopoulos. A Commentary on Teichmüller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen". A. Papadopoulos. Handbook of Teichmüller theory, Volume V, European Mathematical Society Publishing House, pp.569-580, 2016, 978-3-03719-160-6. ⟨10.4171/160-1/13⟩. ⟨hal-01213850⟩