Classical and quantum Teichmüller spaces (Q2854101)

Teichmüller theory has many connections with other directions in mathematical sciences, and is closely related to mathematical physics. In this survey the authors present the main directions of the development of Teichmüller theory and its applications to string theory. In the first chapter they introduces the protagonists of Teichmüller theory: Teichmüller, Ahlfors and Bers. The other two chapters are devoted to the developments related to the Teichmüller space of compact Riemann surfaces and the universal Teichmüller space, respectively, containing basic theory and recent results.NEWLINENEWLINEAfter introducing the definition of the Teichmüller space of finite Riemann surfaces and its tangent and cotangent spaces, the authors are particularly interested in the Kobayashi metric and the Carathéodory metric. They introduce several non-equivalent methods for compactifying the Teichmüller space. In the eighth subsection of Chapter 2, the authors discuss the harmonic properties of a functional called the modulus on the Teichmüller space and describe the Teichmüller metric in terms of it.NEWLINENEWLINEIn Chapter 3, the authors present some main properties of the universal Teichmüller space \(\mathcal{T}\), such as its complex structures, the tangent map of the composite map of the natural projection with the Bers embedding, the Kähler metric on \(\mathcal{T}\). They discuss some subspaces of the universal Teichmüller space, such as the classical Teichmüller space \(T(G)\), the space of normalized diffeomorphisms. In the fifth subsection of Chapter 3, they focus on a Grassmannian realization of the universal Teichmüller space. The last two subsections are devoted to the geometric quantizations of the space of normalized diffeomorphisms and the universal Teichmüller space.