## 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.

### 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)