Chapitre D'ouvrage Année : 2020

On five papers by Herbert Grötzsch

Résumé

Herbert Grötzsch is the main founder of the theory of quasiconformal mappings. In a series of papers written between 1928 and 1932, he introduced these mappings as a natural generalization of conformal mappings and he developed their main properties. He saw that several problems in conformal geometry naturally lead to problems on quasiconformal mappings. In this chapter, we review five of his papers that show the progress of his work from conformal to quasiconformal geometry. The five papers are the following: (1) ``Über einige Extremalprobleme der konformen Abbildung" (On some extremal problems of the conformal mapping), published in 1928; (2) ``Über einige Extremalprobleme der konformen Abbildung. II" (On some extremal problems of the conformal mapping. II), published in 1928; (3) ``Über die Verzerrung bei schlichten nichtkonformen Abbildungen und \"uber eine damit zusammenhängende Erweiterung des Picardschen Satzes." (On the distortion of univalent non-conformal mappings and a related extension of the Picard theorem) \cite{gr3}, published in 1928; (4) ``Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche" (On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions), published in 1930; (5) ``Über möglichst konforme Abbildungen von schlichten Bereichen" (On closest-to-conformal mappings of schlicht domains), published in 1932. Translations of these five papers into English are included in Volume VII of the Handbook. In this paper, we present the main results of these papers, providing introductory remarks, establishing the connections between them, and explaining the background. We shall also indicate some relations with works of Lavrentieff and Teichm\"uller.

Dates et versions

hal-02514898 , version 1 (23-03-2020)

Identifiants

Citer

Vincent Alberge, Athanase Papadopoulos. On five papers by Herbert Grötzsch. Handbook of Teichmüller Theory, Volume VII (ed. A. Papadopoulos), European Mathematical Society Publishing House, Berlin, p. 249-415, 2020, 978-3-03719-203-0. ⟨10.4171/203⟩. ⟨hal-02514898⟩
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