A commentary on Teichmüller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale"

Vincent Alberge (1) , Athanase Papadopoulos (1) , Weixu Su (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Fudan University [Shanghai]

We provide a commentary on Teichmüller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale" (Extremal quasiconformal mappings of closed oriented Riemann surfaces), Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 1940, No.22, 1-197 (1940). The paper is quoted in several works, although it was read by very few people. Some of the results it contains were rediscovered later on and published without any reference to Teichmüller. In this commentary, we highlight the main results and the main ideas contained in that paper and we describe some of the important developments they gave rise to.

Mots clés: deformations of complex structures; Riemann surface; Teichmueller geodesic; Teichmueller space; Teichmueller metric; Teichmueller disc; Riemann moduli problem; moduli; Riemann moduli space; Beltrami differential; Beltrami equation; Fuchsian group; quasiconformal mapping; uniformization; Fricke space

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

Vincent Alberge, Athanase Papadopoulos, Weixu Su. A commentary on Teichmüller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale". A. Papadopoulos. Handbook of Teichmüller theory, Volume V, European Mathematical Society Publishing House, p. 485-531, 2016, ⟨10.4171/160-1/11⟩. ⟨hal-01222188⟩