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**Teichmüller theory and quadratic differentials.**
*(English)*
Zbl 0629.30002

Pure and Applied Mathematics. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. XVII, 236 p.; £43.10 (1987).

The exposition includes the original Teichmüller theory on moduli problems, later results on the complex structure of the Teichmüller space by Ahlfors and Bers, theorems on Teichmüller metrics by Earle, Eells, O’Bryne and Royden, the description of extremal properties for quasiconformal mapping found by Reich and Strebel and Thurston in topological deformation theory for Riemann surfaces.

The Chapter 1 summarizes results from Riemann surface theory and quasiconformal mappings which are selected to provide the necessary background for the Teichmüller theory. The Chapter 2 studies holomorphic quadratic differentials \(\phi\) of finite norm on a Riemann surface R which is compact except for a finite number n of punctures and proves minimal properties for \(\phi\). The Chapters 3 and 4 generalize the inequality of Reich and Strebel to the case when the integration region R is replaced by a fundamental domain \(\omega\) for Fuchsian groups and a relevant space of holomorphic quadratic differential forms is of finite or infinite dimension. The Chapter 5 gives the definition of Teichmüller spaces and of Fuchsian groups and describes under what conditions they are equivalent. It also gives the manifold structure for Teichmüller spaces. The Chapter 6 proves the Teichmüller theorem as well as a necessary and sufficient condition of the Hamilton-Krushkal condition for a quasiconformal mapping to be extremal in its Teichmüller class. The Chapter 7 studies Teichmüller’s and Kobayashi’s metrics. The discontinuity of the action of the modular group is the primary result of the Chapter 8. The Chapter 9 generalizes Royden’s theorem on the coincidence of the Teichmüller modular group with the full group of biholomorphic selfmappings of the Teichmüller space to the case of the surface of finite analytic type. The Chapter 10 clarifies a topological meaning of the dimension \(d=3g-3+n>0\) of the space of integrable holomorphic, quadratic differentials on R of type (g,n). It is the maximal number of simply closed curves on R whose homotopy classes can be represented by nonintersecting curves which are not homotopic to a puncture or to each other. The various properties of the quadratic differentials corresponding to such a family of curves are studied. The Chapter 11 solves an extremal problem of Dirichlet type finding the minimal possible \(L_ 1\)-norm of a continuous quadratic differential on R subject to a certain side condition. Two types of theorems come through all the book - various uniqueness theorems which follow from the length-area principle of Grötzsch and various existence theorems. Besides two additional subfacts - the surjectivity of the Poincaré theta series operator for quadratic differentials and the Ahlfors-Bers density theorem for quadratic differentials are presented.

The Chapter 1 summarizes results from Riemann surface theory and quasiconformal mappings which are selected to provide the necessary background for the Teichmüller theory. The Chapter 2 studies holomorphic quadratic differentials \(\phi\) of finite norm on a Riemann surface R which is compact except for a finite number n of punctures and proves minimal properties for \(\phi\). The Chapters 3 and 4 generalize the inequality of Reich and Strebel to the case when the integration region R is replaced by a fundamental domain \(\omega\) for Fuchsian groups and a relevant space of holomorphic quadratic differential forms is of finite or infinite dimension. The Chapter 5 gives the definition of Teichmüller spaces and of Fuchsian groups and describes under what conditions they are equivalent. It also gives the manifold structure for Teichmüller spaces. The Chapter 6 proves the Teichmüller theorem as well as a necessary and sufficient condition of the Hamilton-Krushkal condition for a quasiconformal mapping to be extremal in its Teichmüller class. The Chapter 7 studies Teichmüller’s and Kobayashi’s metrics. The discontinuity of the action of the modular group is the primary result of the Chapter 8. The Chapter 9 generalizes Royden’s theorem on the coincidence of the Teichmüller modular group with the full group of biholomorphic selfmappings of the Teichmüller space to the case of the surface of finite analytic type. The Chapter 10 clarifies a topological meaning of the dimension \(d=3g-3+n>0\) of the space of integrable holomorphic, quadratic differentials on R of type (g,n). It is the maximal number of simply closed curves on R whose homotopy classes can be represented by nonintersecting curves which are not homotopic to a puncture or to each other. The various properties of the quadratic differentials corresponding to such a family of curves are studied. The Chapter 11 solves an extremal problem of Dirichlet type finding the minimal possible \(L_ 1\)-norm of a continuous quadratic differential on R subject to a certain side condition. Two types of theorems come through all the book - various uniqueness theorems which follow from the length-area principle of Grötzsch and various existence theorems. Besides two additional subfacts - the surjectivity of the Poincaré theta series operator for quadratic differentials and the Ahlfors-Bers density theorem for quadratic differentials are presented.

Reviewer: V.Z.Enolskij