Moduli space, Lecture 4

25 Janeiro 2022, 10:00 • Davide Masoero

We recall how to construct a new complex structure on a Riemann surface starting from a given solution of the Beltrami equation. We give two descriptions of the uniformization of a Riemann surface of genus g>1 with a deformed complex structure. We introduce the notion of Schwarzian derivative, and describe some of its main properties. We introduce a sup-norm on the space of quadratic differentials. We formulate the statement of the Bers Embedding Theorem, we show that the embedding map is well-defined on the Teichmüller space, and that it is injective. Comments on Nehari's and Ahlfors-Weill's Theorems are given.