Riemann surfaces, Teichmüller theory and harmonic maps

A Riemannian metric on an oriented surface, i.e. a two-dimensional oriented real manifold, turns the surface into a one-dimensional complex manifold.

Teichmüller space is a differential geometric approach to the classification of complex structures on the surface. Due to its invariances, namely conformal and diffeomorphism invariance the action functional of harmonic maps on surfaces serves as a tool to study Teichmüller space. After a brief introduction to Riemann surfaces we want to use the theory of harmonic maps to prove

• the uniformization theorem: Every Riemann surface is the quotient of one of the three simply connected Riemann surfaces - Riemann sphere, the complex plane or the complex upper half plane.
• Teichmüller theorem: The Teichmüller space is diffeomorphic to the space of holomorphic quadratic differentials on a given Riemann surface.

Here we will mostly follow the book "Compact Riemann surfaces" by Jürgen Jost. Depending on time and interest of the audience, we want to treat the Weil-Peterson geometry of Teichmüller space or give an outlook to the case of super Riemann surfaces.

Date and time info
Friday 09:30 - 11:00

Keywords
Riemann surfaces, Teichmüller space, harmonic maps

Prerequisites
basic differential geometry

Audience
MSc students, PhD students, Postdocs

Language
English