An Introduction to Teichmüller Spaces

Frontmatter

Chapter 1. Teichmüller Space of Genus g

Abstract

In this chapter, we construct the Teichmüller space T _g of genus g, and give motivations and backgrounds for the following chapters. Some proofs are rather sketchy, and some shall be omitted.

Yoichi Imayoshi, Masahiko Taniguchi

Chapter 2. Fricke Space

Abstract

The purpose of the present chapter is to show that the Teichmüller space of genus g(≧ 2) is realized as a subset in R ^6g−6, which is called the Fricke space of genus g.

Yoichi Imayoshi, Masahiko Taniguchi

Chapter 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

Abstract

In this chapter, we shall discuss some aspects of the hyperbolic geometry on Riemann surfaces which is induced by the Poincaré metric on the unit disk.

Yoichi Imayoshi, Masahiko Taniguchi

Chapter 4. Quasiconformal Mappings

Abstract

In this chapter, we shall explain basic properties of quasiconformal mappings which are needed in the later chapters. In Chapter 1, we have defined differentiable quasiconformal mappings. Here, we weaken the differentiability condition in the definition. Such a technical improvement allows us to use quasiconformal mappings as a tool in considerably more general situations, while formal treatment of such mappings remains the same as in the differentiable case.

Yoichi Imayoshi, Masahiko Taniguchi

Chapter 5. Teichmüller Spaces

Abstract

In this chapter, we shall construct Teichmüller spaces alternatively by using quasiconformal mappings. First, in Section 1, we give a new definition of the Teichmüller space of an arbitrary Riemann surface by using quasiconformal mappings. In Sections 2 and 3, we investigate the case of closed Riemann surfaces of genus g (≥ 2), and prove Teichmüller’s theorem, which states that the Teichmüller space of a closed Riemann surface of genus g (≥ 2) is homeomorphic to the open unit ball in the real (6g – 6)-dimensional Euclidean space. The key of the proof is the existence and uniqueness of the extremal quasiconformal mappings, called Teichmüller mappings.

Yoichi Imayoshi, Masahiko Taniguchi

Chapter 6. Complex Analytic Theory of Teichmüller Spaces

Abstract

We introduce a natural complex manifold structure of the Teichmüller space T(R) of a closed Riemann surface R of genus g (≧ 2), which is realized as a bounded domain in C^3g-3. Furthermore, we prove that the Teichmüller modular group Mod(R) acts properly discontinuously as a group of biholomorphic automorphisms of T(R).

Yoichi Imayoshi, Masahiko Taniguchi

Chapter 7. Weil-Petersson Metric

Abstract

Unless otherwise stated, a Fuchsian group Γ, considered in this chapter, is a Fuchsian model of a closed Riemann surface of genus g (≧ 2). We also assume that each of 0, 1, and ∞ is fixed by an element in Γ — {id}.

Yoichi Imayoshi, Masahiko Taniguchi

Chapter 8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

Abstract

In this chapter, we shall give a beautiful representation, due to S. Wolpert, of the Weil-Petersson fundamental form on T _g (g ≥ 2) by using Fenchel-Nielsen coordinates.

Yoichi Imayoshi, Masahiko Taniguchi

Backmatter