A Commentary on Teichmüller’s paper &lt;i&gt;Ein Verschiebungssatz der quasikonformen Abbildung&lt;/i&gt;

Lipman Bers, a Life in Mathematics

2015

We survey some major contributions to Riemann's moduli space and Teichmüller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space and Teichmüller space. We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory. The period we are interested in starts with Riemann, in 1851, and ends in the early 1960s, when Ahlfors and Bers confirmed that Teichmüller's results were correct.This paper was written for the book Lipman Bers, a life in Mathematics, edited by Linda Keen , Irwin Kra and Rubi Rodriguez (Amercian Mathematical Society, 2015). It is dedicated to the memory of Lipman Bers who was above all a complex analyst and spent a large part of his life and energy working on the analytic structure of Teichmüller space. His work on analysis is nevertheless inseparable from geometry and topology. In this survey, we highlight the relations and the logical dependence between this work and the works of Riemann, Poincaré, Klein, Brouwer, Siegel, Teichmüller, Weil, Grothendieck and others. We explain the motivation behind the ideas. In doing so, we point out several facts which seem to be unknown to many Teichmüller theorists.

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confidence: 99%

On the early history of moduli and Teichmüller spaces

A'Campo

Lipman Bers, a Life in Mathematics

2015

“…These results generalize Koebe's distortion theorems. 6. On the content of Grötzsch's paper [14] The paper [14] is the first which contains Grötzsch's notion of quasiconformal mappings.…”

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confidence: 99%

On five papers by Herbert Grötzsch

Alberge

IRMA Lectures in Mathematics and Theoretical Physics

2020

Self Cite

Herbert Grötzsch is the main founder of the theory of quasiconformal mappings. We review five of his papers, written between 1928 and 1932, that show the progress of his work from conformal to quasiconformal geometry. This will give an idea of his motivation for introducing quasiconformal mappings, of the problems he addressed and on the results he obtained concerning these mappings.The final version of this paper will appear in Vol. VII of the Handbook of Teichmüller theory.

“…This contains at the same time the idea of an almostcomplex structure on the tangent space to Teichmüller space and it is at the basis of the theory of variation of higher-dimensional complex manifolds developed by Kodaira and Spencer. (10) The identification of the tangent space at a point of Teichmüller space with the topological dual of the space of holomorphic quadratic differentials. (11) The introduction and the study of Teichmüller discs (which Teichmüller calls "complex geodesics").…”

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confidence: 99%

“…For a more global view of Teichmüller's contribution on moduli of Riemann surfaces, the reader is also referred to his other papers [39], [37], [41], [32], [34], [40] and the corresponding commentaries [14], [5], [11], [2], [10], [3], as well as the survey papers [1], [19], [4].…”

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confidence: 99%

“…One can also be reminded here of the fact, recalled by Abikoff in [1], that the style of the journal Deutsche Mathematik in which most of Teichmüller's articles appeared was to concentrate on general ideas and not on technical details. This was supposed to be a certain idea of "German mathematics," in the tradition of Riemann and Klein; see [10]. Abikoff in [1] writes: "The tradition of Deutsche Mathematik is one of heuristic argument and contempt for formal proof."…”

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confidence: 99%

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A commentary on Teichmüller’s paper <i>Extremale quasikonforme Abbildungen und quadratische Differentiale</i>

Alberge

IRMA Lectures in Mathematics and Theoretical Physics

2016

Self Cite

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.The final version of this paper, together with the English translation of Teichmüller's paper, will apper in Volume V of the Handbook of Teichmüller theory (European Mathematical Society Publishing House, 2015).1 The word "conjecture," in this setting, sometimes means a claim which is not proved immediately, but which is proved later in the paper. For instance, a conjecture is made in § 100, and § 101 starts with: "This extremely insufficiently grounded conjecture shall now be proved."