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."

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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) [20], (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 ≥ 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 space). 1 After that, he 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 Poincaré and Klein's use of the so-called "continuity principle" in their attempts to prove the uniformization theorem.

The type problem is the problem of deciding, for a simply connected Riemann surface, whether it is conformally equivalent to the complex plane or to the unit dic in the complex plane.We report on Teichmüller's results on the type problem from his two papers Eine Anwendung quasikonformer Abbildungen auf das Typenproblem (An application of quasiconformal mappings to the type problem) (1937) and Untersuchungenüber konforme und quasikonforme Abbildungen (Investigations on conformal and quasiconformal mappings) (1938). They concern simply connected Riemann surfaces defined as branched covers of the sphere. At the same time, we review the theory of line complexes, a combinatorial device used by Teichmüller and others to encode branched coverings of the sphere.In the first paper, Teichmüller proves that any two simply connected Riemann surfaces which are branched coverings of the Riemann sphere with finitely many branch values and which have the same line complex are quasiconformally equivalent. For this purpose, he introduces a technique for piecing together quasiconformal mappings. He also obtains a result on the extension of smooth orientation-preserving diffeomorphisms of the circle to quasiconformal mappings of the disc which are conformal at the boundary.In the second paper, using line complexes, Teichmüller gives a type criterion for a simply-connected surface which is a branched covering of the sphere, in terms of an adequately defined measure of ramification, defined by a limiting process. The result says that if the surface is "sufficiently ramified" (in a sense to be made precise), then it is hyperbolic. In the same paper, Teichmüller answers by the negative a conjecture made by Nevanlinna which states a criterion for parabolicity in terms of the value of a (different) measure of ramification, defined by a limiting process. Teichmüller's results in his first paper are used in the proof of the results of the second one.

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