On five papers by Herbert Grötzsch

Abstract

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.

A Commentary on Teichm{ü}ller's paper "Untersuchungen über konforme und quasikonforme Abbildungen" (Investigations on conformal and quasiconformal mappings) (to appear in Vol. VII of the \emph{Handbook of Teichmüller theory}

“…This kind of module inequalities and of solutions of extremal problems involving ring domains were first studied by Grötzsch in [31]. The reader may also refer to the chapter [12] in the present volume containing an exposition of these results.…”

“…Following Teichmüller, we recall that C µ approaches a circle as µ → ∞ 6 A different and shorter proof of a variation of Theorem 4.1, together with an estimate equivalent to (13), is due to Pommerenke [52], p. 201-202. The result proved by Pommerenke uses the additional assumption that G ′ ∪ G ′′ = G, and thus the estimate (12) concerns the points lying on the joint boundary components ∂G ′ ∩ ∂G ′′ . Pommerenke's proof uses the method of extremal length, properties of univalent conformal mappings, the area theorem, and coefficient estimates.…”

Quasiconformal mappings and curvatures on metric measure spaces

“…Originating in cartography that represents the regions of the surface of the earth on a Euclidean piece of paper and beginning in the works of Tissot [42,44,50], Grötzsch [4,25], Lavrentieff [3,33] and others [43], the study of quasiconformal mappings on the Euclidean spaces E n has a rich history of over one hundred years, see the books [2,21] and the references therein.…”