##
**The complex analytic theory of Teichmüller spaces.**
*(English)*
Zbl 0667.30040

Canadian Mathematical Society Series of Monographs and Advanced Texts. New York etc.: Wiley. xiii, 427 p. £47.50 (1988).

The book of S. Nag markes a new stage in the intensive development of the Teichmüller space theory. Indeed, it offers for the first time a systematic access way till the core of the most recent results in the complex analytic theory of these and other moduli spaces. It represents not only a very necessary but also very important achievement providing a self-contained, clear and attractive exposition in spite of the huge amount of concepts and facts which are presented of the complexity of the proofs and the variety of the utilized technics. To this essentially contributed author’s manner to point out the principal aspects of the theory and the main ideas of the proofs, the comments and remarks scattered throughout the book, the numerous representative examples and the exercises which assure the active study.

After a preface and an introduction, which initiate the reader in the aim, the history and the principal topics of the theory, Chapter 1 presents auxiliary knowledge on Riemann surfaces and their uniformizing groups, quasiconformal mappings, automorphic forms, unconventional surfaces topology (Nielsen theorem, the mapping class groups), the infinite-dimensional holomorphy, and a résumé of results on Riemann surfaces (vector bundles, Riemann-Roch theorem, Jacobi’s variety and Abel’s theorem, connections with algebraic curves, conformal structures).

Chapter 2 defines different moduli spaces: the Teichmüller space T(X) and the reduced one \(T^{\#}(X)\), the Riemann space R(X), the Torelli, Earle and Kra-Nag spaces of the Riemann surface X as well as for the Fuchsian group G: T(G), \(T^{\#}(G)\) and so on, by classes of quasiconformal mappings or of Beltrami differentials in \(L^{\infty}_{(-1,1)}(X)_ 1=L^{\infty}(U,G)_ 1.\) It deals with the Teichmüller metric, the modular groups and the representation of the Teichmüller space as an orbit space. The finite-dimensional case of Teichmüller spaces is then treated in detail. By using Fricke coordinates, a real analytic structure is introduced on \(T(g,n)=T(X)\) for X of the finite conformal type (g,n), \(2g-2+n>0\), and on \(T^{\#}(g,n,m)=T^{\#}(X)\) for X of finite topological type (g,n,m). Both spaces are represented as an open domain in \({\mathbb{R}}^ N\), \(N=6g- 6+2n\) or \(6g-6+2n+3m\), respectively. A nice exposition of Teichmüller’s theorem clarifies the role played by the Teichmüller extremal maps. As a consequence T(g,n) and \(T^{\#}(g,n,m)\) embed as an open ball in \({\mathbb{R}}^ N\). Further the proper discontinuity of the action of the modular groups on finite-dimensional Teichmüller spaces is established and used to deduce the complex structure on R(X) and on other intermediate spaces.

The central result of the book forms the content of Chapter 3:

T(X) and T(G) carry a unique structure of complex Banach manifold. In the finite-dimensional case, this “canonical complex structure” is constructed by means of the fundamental projection \(\Phi: L^{\infty}_{(-1,1)}(X)_ 1\to T(X)\) and the Fricke homeomorphic embedding F: T(X)\(\to Im F\subset {\mathbb{R}}^ N\). A unique complex structure is given on Im F such that \(\tilde F=F\circ \Phi\) becomes a holomorphic submersion and this structure is pulled back onto T(X) via F. Thus T(g,n) becomes a \((3g-3+n)\)-dimensional complex analytic manifold. In the general case, T(G) is characterized by \(w^{\mu}\), where \(\mu \in L^{\infty}(U,G)_ 1\) is extended with 0 to L, the Bers projection \(\Phi_{\beta}: L^{\infty}(U,G)_ 1\to B_ 2(L,G)\) and the Bers embedding \(\beta: T(G)\to B_ 2(L,G),\) \(\Phi_{\beta}=\beta \circ \Phi\), replacing \(\tilde F\) and F. Author proves that \(\Phi_{\beta}\) is a holomorphic submersion onto its image \(T_{\beta}(G)\), an open domain in \(B_ 2(L,G)\), and T(G) becomes via \(\beta\) a complex analytic Banach manifold. Results about \(\partial T_{\beta}(G)\) in \(T_{\beta}(1)\), the existence of the Ahlfors-Weill local sections for \(\Phi_{\beta}\) and Earle’s embedding of \(T^{\#}(G)\) via \(\beta\) are presented. The study of the complex analytic structure of the Teichmüller spaces is continued in Chapter 4 by explicit local moduli for T(g,n): periods of Abelian differentials (Ahlfors, Rauch, Earle), stratification moduli for quasi-Fuchsian groups (Kra-Maskit, Earle), parameters for Schiffer variation on the underlying Riemann surfaces (Patt, Gardiner, Nag). Further \(T_{\beta}(G)\) is interpreted as a subset of the space of deformations of G and as a subset of projective structures on L/G, and its boundary in \(B_ 2(L,G)\) is thoroughly studied in connection with the totally degenerated Kleinian groups (Bers).

Chapter 5 is concerned with the Bers fiber space F(G) over T(G), when G is a Fuchsian group, and the tautological fiber space V(G) over T(G) deduced as a quotient of F(G) with respect to the action of G (G torsion- free). Author proves, by using the Earle-Eells fiber bundle description of Teichmüller spaces, the Bers isomorphism between F(g,n) and \(T(g,n+1)\) and constructs the space of varying projective structures on varying Riemann surfaces. The marked n-pointed families of compact genus g Riemann surfaces are introduced as holomorphic families of such surfaces over a complex Banach manifold. Author proves that the universal family V(g,n) over T(g,n) is the universal object in the category of these marked n-pointed families, whence he deduces the axiomatic characterization of T(g,n) as a universal object (Grothendieck, Earle, Engber). An Appendix is dedicated to invariant metrics on moduli spaces (Teichmüller, Kobayashi, Carathéodory, Weil-Petersson, Thurston and Bergman metrics). Finally, connections with the hyperbolic and the straight-space geometry, as well as other important theorems (Hamilton, Bers-Greenberg, Royden, Earle-Kra, Wolpert) are presented. By its rich information, by the new points of view due to the author research work, by its lucid and very rigorous style, the book constitutes an interesting and remarkable work for graduate students and experts in Teichmüller theory, as well as for physicists which study string theory.

After a preface and an introduction, which initiate the reader in the aim, the history and the principal topics of the theory, Chapter 1 presents auxiliary knowledge on Riemann surfaces and their uniformizing groups, quasiconformal mappings, automorphic forms, unconventional surfaces topology (Nielsen theorem, the mapping class groups), the infinite-dimensional holomorphy, and a résumé of results on Riemann surfaces (vector bundles, Riemann-Roch theorem, Jacobi’s variety and Abel’s theorem, connections with algebraic curves, conformal structures).

Chapter 2 defines different moduli spaces: the Teichmüller space T(X) and the reduced one \(T^{\#}(X)\), the Riemann space R(X), the Torelli, Earle and Kra-Nag spaces of the Riemann surface X as well as for the Fuchsian group G: T(G), \(T^{\#}(G)\) and so on, by classes of quasiconformal mappings or of Beltrami differentials in \(L^{\infty}_{(-1,1)}(X)_ 1=L^{\infty}(U,G)_ 1.\) It deals with the Teichmüller metric, the modular groups and the representation of the Teichmüller space as an orbit space. The finite-dimensional case of Teichmüller spaces is then treated in detail. By using Fricke coordinates, a real analytic structure is introduced on \(T(g,n)=T(X)\) for X of the finite conformal type (g,n), \(2g-2+n>0\), and on \(T^{\#}(g,n,m)=T^{\#}(X)\) for X of finite topological type (g,n,m). Both spaces are represented as an open domain in \({\mathbb{R}}^ N\), \(N=6g- 6+2n\) or \(6g-6+2n+3m\), respectively. A nice exposition of Teichmüller’s theorem clarifies the role played by the Teichmüller extremal maps. As a consequence T(g,n) and \(T^{\#}(g,n,m)\) embed as an open ball in \({\mathbb{R}}^ N\). Further the proper discontinuity of the action of the modular groups on finite-dimensional Teichmüller spaces is established and used to deduce the complex structure on R(X) and on other intermediate spaces.

The central result of the book forms the content of Chapter 3:

T(X) and T(G) carry a unique structure of complex Banach manifold. In the finite-dimensional case, this “canonical complex structure” is constructed by means of the fundamental projection \(\Phi: L^{\infty}_{(-1,1)}(X)_ 1\to T(X)\) and the Fricke homeomorphic embedding F: T(X)\(\to Im F\subset {\mathbb{R}}^ N\). A unique complex structure is given on Im F such that \(\tilde F=F\circ \Phi\) becomes a holomorphic submersion and this structure is pulled back onto T(X) via F. Thus T(g,n) becomes a \((3g-3+n)\)-dimensional complex analytic manifold. In the general case, T(G) is characterized by \(w^{\mu}\), where \(\mu \in L^{\infty}(U,G)_ 1\) is extended with 0 to L, the Bers projection \(\Phi_{\beta}: L^{\infty}(U,G)_ 1\to B_ 2(L,G)\) and the Bers embedding \(\beta: T(G)\to B_ 2(L,G),\) \(\Phi_{\beta}=\beta \circ \Phi\), replacing \(\tilde F\) and F. Author proves that \(\Phi_{\beta}\) is a holomorphic submersion onto its image \(T_{\beta}(G)\), an open domain in \(B_ 2(L,G)\), and T(G) becomes via \(\beta\) a complex analytic Banach manifold. Results about \(\partial T_{\beta}(G)\) in \(T_{\beta}(1)\), the existence of the Ahlfors-Weill local sections for \(\Phi_{\beta}\) and Earle’s embedding of \(T^{\#}(G)\) via \(\beta\) are presented. The study of the complex analytic structure of the Teichmüller spaces is continued in Chapter 4 by explicit local moduli for T(g,n): periods of Abelian differentials (Ahlfors, Rauch, Earle), stratification moduli for quasi-Fuchsian groups (Kra-Maskit, Earle), parameters for Schiffer variation on the underlying Riemann surfaces (Patt, Gardiner, Nag). Further \(T_{\beta}(G)\) is interpreted as a subset of the space of deformations of G and as a subset of projective structures on L/G, and its boundary in \(B_ 2(L,G)\) is thoroughly studied in connection with the totally degenerated Kleinian groups (Bers).

Chapter 5 is concerned with the Bers fiber space F(G) over T(G), when G is a Fuchsian group, and the tautological fiber space V(G) over T(G) deduced as a quotient of F(G) with respect to the action of G (G torsion- free). Author proves, by using the Earle-Eells fiber bundle description of Teichmüller spaces, the Bers isomorphism between F(g,n) and \(T(g,n+1)\) and constructs the space of varying projective structures on varying Riemann surfaces. The marked n-pointed families of compact genus g Riemann surfaces are introduced as holomorphic families of such surfaces over a complex Banach manifold. Author proves that the universal family V(g,n) over T(g,n) is the universal object in the category of these marked n-pointed families, whence he deduces the axiomatic characterization of T(g,n) as a universal object (Grothendieck, Earle, Engber). An Appendix is dedicated to invariant metrics on moduli spaces (Teichmüller, Kobayashi, Carathéodory, Weil-Petersson, Thurston and Bergman metrics). Finally, connections with the hyperbolic and the straight-space geometry, as well as other important theorems (Hamilton, Bers-Greenberg, Royden, Earle-Kra, Wolpert) are presented. By its rich information, by the new points of view due to the author research work, by its lucid and very rigorous style, the book constitutes an interesting and remarkable work for graduate students and experts in Teichmüller theory, as well as for physicists which study string theory.

Reviewer: C.Andreian Cazacu

### MSC:

30F99 | Riemann surfaces |

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

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

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |