On Teichmüller's proof of Teichmüller's theorem
The author gives further explanations of his proof of Teichmüller's theorem, first presented in his expository paper in Princeton, Math. Ser. 24, 84-119 (1960; Zbl 0100.289). A key element in the proof was the assertion that parameters determining a marked Fuchsian group depend continuously on the parameters determining the Teichmüller Beltrami coefficient. From Bers' exposition it appeared to some readers that the theory of quasiconformal mappings with bounded measurable Beltrami coefficients was indispensable to the proof of Teichmüller's theorem. Not being able to comprehend Teichmüller's arguments, some of them erroneously concluded that Teichmüller had given an incomplete proof. In this paper, Bers shows that the continuity can be proved in an elementary way which does not involve the complete theory of quasiconformal mappings and, in fact, is a way very close to the one employed by Teichmüller. The route taken is to use Grötzsch's inequality bounding the distortion of the modulus of a rectangle or an annulus in terms of the average dilatation of a quasiconformal mapping. Then, in order to get the continuous dependence of the parameters determining a marked Fuchsian group, one only needs to show the continuous dependence of the values of a quasiconformal mapping restricted to the real axis. The values on the real axis can be seen as depending on the moduli of rectangles with vertices on the real axis. Then one shows the convergence of the moduli of rectangles by showing that the local dilatations are bounded functions and showing that they converge uniformly on compact subsets.
scientific article; zbMATH DE number 3979466