The Complex Analytic Theory of Teichmuller SpacesAn accessible, self-contained treatment of the complex structure of the Teichmüller moduli spaces of Riemann surfaces. Complex analysts, geometers, and especially string theorists (!) will find this work indispensable. The Teichmüller space, parametrizing all the various complex structures on a given surface, itself carries (in a completely natural way) the complex structure of a finite- or infinite-dimensional complex manifold. Nag emphasizes the Bers embedding of Teichmüller spaces and deals with various types of complex-analytic coördinates for them. This is the first book in which a complete exposition is given of the most basic fact that the Bers projection from Beltrami differentials onto Teichmüller space is a complex analytic submersion. The fundamental universal property enjoyed by Teichmüller space is given two proofs and the Bers complex boundary is examined to the point where totally degenerate Kleinian groups make their spectacular appearance. Contains much material previously unpublished. |
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Contents
Introduction | 1 |
A Portmanteau of Preliminaries | 8 |
A Résumé of Results | 89 |
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Ahlfors arbitrary Fuchsian group B₁ B₂(L Banach manifold Banach space Beltrami differentials Bers projection biholomorphic automorphism boundary bundle canonical coefficients compact Riemann surface complex analytic complex Banach complex dilatation complex manifold complex structure convergence corresponding course D₁ D₂ defined definition diffeomorphism disk domain dx dy equation fact fiber space finite conformal type finite-dimensional fixed points Fuchsian group G₁ genus g group G holomorphic function holomorphic map holomorphic submersion homeomorphism homotopy hyperbolic identity induced integral isometry isomorphism Kleinian groups linear Möb(C Möb(R Möbius transformation mod G modular group moduli space morphism normalized Note parameters Poincaré precisely proof Proposition prove punctures quadratic differentials quasicircle quasiconformal homeomorphism quasiconformal mappings quotient real-analytic Remark Riemannian schlicht subgroup subset subspace tangent Teichmüller metric Teichmüller space Teichmüller's lemma theorem topological torsion-free Fuchsian group trivial type g universal covering vector w₁ X₁ zero