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**An introduction to Teichmüller spaces.**
*(English)*
Zbl 0754.30001

Tokyo: Springer-Verlag. xii, 279 p. (1992).

An excellent, well written survey of the theory of Teichmüller Spaces, filling an often deplored gap in mathematical literature. Certainly a ‘must’ for everyone working in the field. We give a brief description of the contents:

Chapter 1 gives primitive motivations and backgrounds for the following chapters. The Teichmüller space \(T_ g\) is constructed as the set of marked Riemann surfaces of genus \(g\).

Chapter 2 deals with the approach using Fuchsian groups \(\Gamma\). Marked Riemann surfaces are identified with the corresponding systems of generators of \(\Gamma\). \(T_ g\) is thereby represented as a subset of \(\mathbb{R}^{6g-6}\).

In chapter 3 the Fenchel-Nielsen coordinates on \(T_ g\) are defined using length functions and twist parameters of \(3g-3\) simple closed geodesics in the Poincaré metric.

In chapters 4 and 5 the Teichmüller space \(T(R)\) of a closed Riemann surface \(R\) is constructed using quasiconformal mappings. It is shown that \(T_ g\) is homeomorphic to the space \(A_ 2(R)\) of holomorphic quadratic differentials on \(R\) and hence homeomorphic to \(\mathbb{R}^{6g- 6}\).

In chapter 6 the Bers embedding of \(T(R)\) into a bounded domain in \(A_ 2(R^*)\), the space of holomorphic quadratic differentials on the mirror image \(R^*\) of \(R\) is constructed. Using this embedding one sees that the Teichmüller and moduli spaces are normal complex analytic spaces of dimension \(3g-3\).

Chapters 7 and 8 treat the Weil-Petersson metric on \(T_ g\). It is shown that it is Kählerian and its representation in the Fenchel-Nielsen coordinates due to Wolpert is derived.

Two appendices deal with Schiffer’s interior variation and the compactification of moduli spaces. The book also contains almost 20 pages of references.

Chapter 1 gives primitive motivations and backgrounds for the following chapters. The Teichmüller space \(T_ g\) is constructed as the set of marked Riemann surfaces of genus \(g\).

Chapter 2 deals with the approach using Fuchsian groups \(\Gamma\). Marked Riemann surfaces are identified with the corresponding systems of generators of \(\Gamma\). \(T_ g\) is thereby represented as a subset of \(\mathbb{R}^{6g-6}\).

In chapter 3 the Fenchel-Nielsen coordinates on \(T_ g\) are defined using length functions and twist parameters of \(3g-3\) simple closed geodesics in the Poincaré metric.

In chapters 4 and 5 the Teichmüller space \(T(R)\) of a closed Riemann surface \(R\) is constructed using quasiconformal mappings. It is shown that \(T_ g\) is homeomorphic to the space \(A_ 2(R)\) of holomorphic quadratic differentials on \(R\) and hence homeomorphic to \(\mathbb{R}^{6g- 6}\).

In chapter 6 the Bers embedding of \(T(R)\) into a bounded domain in \(A_ 2(R^*)\), the space of holomorphic quadratic differentials on the mirror image \(R^*\) of \(R\) is constructed. Using this embedding one sees that the Teichmüller and moduli spaces are normal complex analytic spaces of dimension \(3g-3\).

Chapters 7 and 8 treat the Weil-Petersson metric on \(T_ g\). It is shown that it is Kählerian and its representation in the Fenchel-Nielsen coordinates due to Wolpert is derived.

Two appendices deal with Schiffer’s interior variation and the compactification of moduli spaces. The book also contains almost 20 pages of references.

Reviewer: S.Timmann (Hannover)

### MSC:

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

30F60 | Teichmüller theory for Riemann surfaces |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |