Moduli space, Lecture 2: Complex and conformal structures, the Beltrami equation

18 Janeiro 2022, 10:00 • Davide Masoero

We introduce the presentation of M_g as a quotient of the Teichmüller space T_g by the Mapping Class Group Gamma_g. We briefly discuss the identification of Gamma_g with the set of path-connected components of the orientation preserving diffeomorphisms group. We show a 1-1 correspondence between complex structures (compatible with a given orientation) and conformal structures on a Riemann surface. We show that finding isothermal coordinates is equivalent to solving the Beltrami equation. Basic properties of the Beltrami equation and its solutions are given. We formulate the statement of the Measurable Mapping Theorem.