TeichmUller's work on the type problem (To appear in Vol. VII of the Handbook of Teichm{\"u}ller theory, 2020)

Abstract

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 Teichm{\"u}ller's results on the type problem from his two papers Eine Anwendung quasikonformer Abbildungen auf das Typenproblem (An application of quasiconformal map-pings to the type problem) (1937) and Untersuchungen{\"u}ber kon-forme und quasikonforme Abbildungen (Investigations on con-formal 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 Teichm{\"u}ller and others to encode branched coverings of the sphere. In the first paper, Teichm{\"u}ller 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 quasi-conformal 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, Teichm{\"u}ller 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, Te-ichm{\"u}ller answers by the negative a conjecture made by Nevan-linna which states a criterion for parabolicity in terms of the value of a (different) measure of ramification, defined by a limiting process. Teichm{\"u}ller's results in his first paper are used in the proof of the results of the second one.The final version of this paper will appear in Vol. VII of the Handbook of Teichm{\"u}ller theory (European Mathematical Society Publishing House, 2020).