A Commentary on Teichmüller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen" - Archive ouverte HAL Accéder directement au contenu
Chapitre D'ouvrage Année : 2015

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

Résumé

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) \cite{T29}, (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 $g\geq 2$. 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\index{Fricke space} space). The consideration of such a space of Fuchsian groups, with the definition of a topology on that space, is already contained in the papers of Poincar\'e and Klein, and the references there to the papers by Poincar\'e and Klein. The Fricke space has acquired in the recent literature a new status, as a subspace of the character variety of the representation space of the fundamental group of the surface in the Lie group $\mathrm{PSL}(2,\mathbb{R})$. After that, Teichmüller 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 Poincar\'e and Klein's use of the so-called ``continuity principle" in their attempts to prove the uniformization theorem.

Dates et versions

hal-01399591 , version 1 (19-11-2016)

Identifiants

Citer

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". Handbook of Teichmüller theory, Vol. V. (A. Papadopoulos, ed.), European Mathematical Society Publishing House, Zurich, 2015, p. 569--580., 2015, 978-3-03719-160-6. ⟨10.4171/160-1/13⟩. ⟨hal-01399591⟩
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