Sumários
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.
Moduli space, Lecture 3
21 Janeiro 2022, 10:00 • Davide Masoero
We study the solvability of the Beltrami equation in case of coefficients with different types of regularity (smooth, measurable and essentially bounded). We prove the existence and uniqueness of a solution in the Sobolev space W^{1,2}_loc(C) in the case of Beltrami essentially bounded coefficients with compact support. Partial results are given, in order to improve the regularity class of this solution in case of smooth coefficients.
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.
Moduli space, Lecture 1: Automorphisms groups of Riemann surfaces, low genus examples of moduli spaces
14 Janeiro 2022, 10:00 • Davide Masoero
After introducing the moduli spaces M_{g,n} of pointed Riemann surfaces as sets, we formulate the main problem of the lectures: to put a geometry on M_{g,n}. We introduce and discuss basic properties of automorphisms groups of Riemann surfaces: we compute their dimensions for all possible values of g, and we prove that such a group is finite if g>1 (Schwarz Theorem). We discuss low genus examples of moduli spaces: M_{0,n} for n≥0, M_1, M_{1,1}.