A commentary on Lavrentieff’s paper "Sur une classe de représentations continues"

Vincent Alberge (1) , Athanase Papadopoulos (2)

1 Fordham University [New York]
2 IRMA - Institut de Recherche Mathématique Avancée

We report on Lavrentieff's paper ``Sur une classe de repr\'esentations continues'' (On a class of continuous representations), published in 1935, in which he introduced quasiconformal mappings under the name ``almost analytic functions." He set the bases of this theory, obtaining several results that show that these functions constitute natural generalizations of conformal mappings. The results include an existence and uniqueness theorem which constitutes an early version of the measurable Riemann mapping theorem, a result on the existence of isothermal coordinates, a generalization of Picard's big theorem to the setting of almost analytic functions, and several results on the type problem.

Mots clés: generalization of Picard's theorem; isothermal coordinates; measurable Riemann mapping theorem; the type problem.; quasiconformal mapping; conformal representation; almost analytic function; characteristic function

Domaines: Topologie géométrique [math.GT]; Histoire et perspectives sur les mathématiques [math.HO]; Variables complexes [math.CV]

Vincent Alberge, Athanase Papadopoulos. A commentary on Lavrentieff’s paper "Sur une classe de représentations continues". Handbook of Teichmüller Theory, Volume VII (ed. A. Papadopoulos), European Mathematical Society Publishing House, Berlin, p. 441–451, 2020, 978-3-03719-203-0. ⟨10.4171/203⟩. ⟨hal-02521316⟩