Teichmüller theory (Q2848312)

These notes summarize a set of five lectures given by the author at the 2011 Park City Mathematics Institute program on moduli spaces of Riemann surfaces. In the context of this thematic program, the goal of these notes is to give an introduction to some of the recent developments in Teichmüller theory, with a particular focus on new tools and on new geometric aspects likewise.NEWLINENEWLINE Lecture 1 introduces the Teichmüller space \(T(S)\) of a closed oriented surface \(S\) of genus \(g\geq 2\) as the space of all marked hyperbolic structures on \(S\). From this viewpoint of hyperbolic surfaces, the classical Fenchel-Nielsen coordinates on \(T(S)\) are explained, and the resulting real analytic manifold structure of \(T(S)\) is thoroughly described in the course of this discussion.NEWLINENEWLINE Lecture 2 turns to some classical analytic aspects of Teichmüller theory. More precisely, the author introduces quasiconformal maps of Riemann surfaces, the spaces of Abelian and quadratic differentials of a Riemann surface, Teichmüller geodesics in \(T(S)\), the Teichmüller metric, and finally discusses the famous Teichmüller existence and uniqueness theorem. The latter states that any two points in Teichmüller space can be connected by a unique Teichmüller geodesic.NEWLINENEWLINE Lecture 3 provides a differential-geometric look at the complex geometry of Teichmüller spaces and moduli spaces of Riemann surfaces, respectively. The topics of this section include Riemann's period relations for compact Riemann surfaces, the Hodge bundle on Teichmüller space, Jacobians of Riemann surfaces and the Torelli map, the Weil-Petersson metric on Teichmüller space, the Kobayashi pseudometric on \(T(S)\) and complex geodesics (or Teichmüller discs) in Teichmüller space.NEWLINENEWLINE Lecture 4 introduces the so-called augmented Teichmüller space \(\overline T(S)\) of a surface \(S\), describes the action of the mapping class group \(\text{Mod}(S)\) on \(\overline T(S)\), and gives then a differential-geometric description of the compact space \(\overline T(S)/\text{Mod}(S)\), which may be regarded as a complex analytic model for the algebraic Deligne-Mumford compactification \(\overline M_g\) of the moduli space \(M_g= T(S)/\text{Mod}(S)\) of compact Riemann surfaces of genus \(g\geq 2\). In this context, some recent research contributions of the author herself are explained.NEWLINENEWLINE Lecture 5, the final part of the present survey article, continues the differential-geometric study of the moduli spaces \(M_g\) with a view toward Teichmüller dynamics. Its main goal is to relate some geometric properties of these moduli spaces to the so-called Teichmüller flow on spaces of quadratic differentials, thereby surveying the recent work of Minsky-Weiss, Smillie-Weiss, Lindenstrauss-Mirzakhani, K. Rafi, and (again) the author herself. This lecture contains much more recent material than the previous sections, and the presentation is also significantly more detailed.NEWLINENEWLINE All together, these notes provide a highly enlightening panoramic view of various aspects of both classical and modern Teichmüller theory, in particular with regard to the interrelation with the moduli theory of Riemann surfaces.NEWLINENEWLINEFor the entire collection see [Zbl 1272.30002].