Foreword
Contents
Introduction to Teichmüller theory, old and new VII
Part A. Surveys
1. The Deligne–Mumford compactification and crystallographic groups
1. Introduction
2. Orbifolds and curve complexes
2.1. Orbifolds
2.2. Curve complexes
3. Augmented Teichmüller spaces
4. The ε-thick part
5. The compactness theorem
6. Construction of the orbifold-charts
7. Crystallographic groups
References
2. Complex geometry of Teichmüller domains
1. Introduction
2. Preliminaries
3. The Bers embedding
3.1. Construction of the embedding
3.2. Structure of the Bers boundary
4. Invariant metrics on Teichmüller domains
4.1. The Kobayashi and Carathéodory metrics
4.2. Complex geodesics
4.3. The Teichmüller metric
5. Comparison with products and bounded symmetric domains
6. Convexity and pseudoconvexity
7. Kobayashi ≠ Carathéodory on Teichmüller domains
8. Local convexity of Teichmüller domains
References
3. Holomorphic quadratic differentials in Teichmüller theory
1. Introduction
2. The co-tangent space of Teichmüller space
3. Singular-flat geometry
4. Extremal maps and Teichmüller geodesics
5. Hopf differentials of harmonic maps
6. The Hubbard–Masur theorem
7. Schwarzian derivative and projective structures
8. Meromorphic quadratic differentials
8.1. Simple poles
8.2. Poles of order two
8.3. Higher order poles
8.4. Meromorphic projective structures
References
4. Mostow strong rigidity of locally symmetric spaces revisited
1. Introduction
2. Deformation and rigidity of Riemann surfaces
3. Deformation and three types of rigidity of complex manifolds
4. Deformation and local rigidity of locally symmetric spaces
5. Strong rigidity of lattices and locally symmetric spaces
6. Rigidity and arithmeticity
6.1. Local rigidity of Hermitian locally symmetric spaces and fields of definition
6.2. Superrigidity and arithmeticity of lattices
7. Proofs of Mostow strong rigidity
8. Quasiconformal mappings and Mostow strong rigidity for hyperbolic manifolds
9. The negative part of Mostow strong rigidity
10. Generalizations of the Mostow strong rigidity
10.1. Strong rigidity for lattices of p-adic semisimple Lie groups
10.2. Generalizations to infinite volume locally symmetric spaces
10.3. Rigidity results in differential geometry
11. Other major works of Mostow and another strong rigidity
12. Some comments on the Mostow strong rigidity
References
5. Models of ends of hyperbolic 3-manifolds. A survey
1. Introduction: Fuchsian surface groups
2. Kleinian surface groups
2.1. Quasi-Fuchsian groups
2.2. Laminations and pleated surfaces
2.3. Degenerate groups
3. Building blocks and model geometries
3.1. Bounded geometry
3.2. i-bounded geometry
3.3. Amalgamation geometry
4. Hierarchiesand the ending lamination theorem
4.1. Hierarchies
5. Split geometry
5.1. Split level surfaces
5.2. Split surfaces and weak split geometry
References
6. Universal Teichmüller space as a non-trivial example of infinite-dimensional complex manifolds
1. Introduction
2. Definition of the universal Teichmüller space
2.1. Quasiconformal maps
2.2. Universal Teichmüller space
2.3. The complex structureof the universal Teichmüller space
3. Subspaces of the universal Teichmüller space
3.1. Classical Teichmüller spaces
3.2. The space of normalized diffeomorphisms
4. Grassmann realization of universal Teichmüller space
4.1. Sobolev space of half-differentiable functions
4.2. Grassmann realization of calT
5. Universal Teichmüller space and string theory
5.1. Classical system associated with smooth string theory
5.2. Quantization of classical system (Ω_d,calA_d)
5.3. Half-differentiable strings
5.4. Quantization of the Sobolev space V_d
References
7. Generalized conformal barycentric extensionsof circle maps
1. Introduction
2. Generalized conformal barycentric extensions of continuous circle maps
2.1. Definition
2.2. Conformal naturalityand anticonformal naturality of the extension
2.3. Continuity of the extension
2.4. Criterion for surjectivity
3. Qualitative and quantitative properties of the extensions of circle homeomorphisms in different classes
3.1. Extensions of homeomorphisms
3.2. Extensions of quasisymmetric homeomorphisms
3.3. Extensions of locally quasisymmetric homeomorphisms
3.4. Extensions of diffeomorphisms
References
8. Higgs bundles and higher Teichmüller spaces
1. Introduction
2. Higgs bundles
2.1. Basic definitions
2.2. Stability of G-Higgs bundles
2.3. Higgs bundles and representations
3. A basic example: G=SL(2,R)
4. The Hitchin map, split real forms and Hitchin components
4.1. Maximal split subgroup and Chevalley map
4.2. The Hitchin map
4.3. Hitchin components for SL(n,R)-Higgs bundles
4.4. The Hitchin–Kostant–Rallis section
5. Hermitian groups and maximal Toledo components
5.1. Hermitian symmetric spaces and Cayley transform
5.2. The Toledo character
5.3. Toledo invariant and Milnor–Wood inequality
5.4. Hermitian groups of tube type and Cayley correspondence
6. SO(p,q)-Higgs bundles and higher Teichmüller spaces
6.1. SO(p,q)-Higgs bundles and topological invariants
6.2. The case 2
6.3. The case q=p+1
6.4. The case q=p
6.5. The case p=2
6.6. Cayley correspondence
7. Positivity and Cayley correspondence
7.1. Anosov and positive representations
7.2. General Cayley correspondence
8. Appendix: tables
References
Part B. Essays on the early works on quasiconformal mappings
9. A note on Nicolas-Auguste Tissot: at the origin of quasiconformal mappings
1. Introduction
2. Biographical note on Tissot
3. On the work of Tissot on geographical maps
References
10. Memories of Herbert Grötzsch
1. Biographical notes
2. Notes on the work
3. Grötzsch and Koebe
4. Spatial almost conformal (extremal quasiconformal) mappings
References
11. A note about Mikhaïl Lavrentieff and his world of analysis in the Soviet Union
1. Introduction
2. Family
3. Luzin
4. Back to Lavrentieff: his mathematics
5. Siberia
6. Mechanics and engineering
7. Appendix. Some clarifications on the history of Russian mathematicsof the 20th century (G. Sinkevich)
References
Part C. Sources
12. A letter
13. On some extremal problems of the conformal mapping
I. Two lemmas
II. Application of the two lemmas
III. The problem of the closest boundary point for domains K_r and arc(K_r)
IV. The problem of the closest boundary point for domains K_0 and arc(K_0)
14. On some extremal problems of the conformal mapping II
V. The exact bounds for |f(z)| in the case of normalized schlicht mappings from R_r and R_0
VI. The exact bounds for |f'(z)| for normalized schlicht mapping of R_r and R_0
15. On the distortion of schlicht non-conformal mappings and on a related extension of Picard's theorem
I. A notion of planar A_Q mapping
II. Properties of the schlicht A_Q
III. The big Picard theorem for mappings of bounded infinitesimal distortion A_Q
16. On the distortion of non-conformal schlicht mappingsof multiply-connected schlicht regions
I. Concept of a mapping A_Q
II. Normalized mappings A_Q
III. Theorems about normalized mappings A_Q
IV. Theorems about mappings A_Q normalized differently
17. On closest-to-conformal mappings
18. On five papers by Herbert Grötzsch
1. Introduction
2. Conformal representations
3. Two lemmas from the paper “Über einige Extremalprobleme der konformen Abbildung”
4. Four theorems from the paper ``Über einige Extremalprobleme der konformen Abbildung”
5. Two theorems from the paper “Über einige Extremalprobleme der konformen Abbildung. II.”
6. On the content of Grötzsch's paper “Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhaängende Erweiterungdes Picardschen Satzes”
7. Some comments on Grötzsch's paper “Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhaängende Erweiterung des Picardschen Satzes”
8. The paper “Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche”
9. The paper “Über möglichst konforme Abbildungenvon schlichten Bereichen”
10. Examples of closest-to-conformal mappings (extremal quasiconformal mappings)
References
19. On a class of continuous representations
1. Preliminary propositions
2. Existence theorems
3. Analytical applications
4. Geometrical applications
20. A commentary on Lavrentieff's paper “Sur une classe de représentations continues”
1. Introduction
2. Almost analytic functions
3. Applications
References
21. An application of quasiconformal mappings to the type problem
1. The problem
2. The dilatation quotient
3. Quasiconformal mappings
4. Surfaces with the same line complex
5. Invariance of the type
6. Control of L on the characteristic
22. Investigations on conformal and quasiconformal mappings
1. The module of a ring domain
2. Distortion theorems
3. The quadrilateral
4. The Modulsatz
5. Taking the limit
6. Quasiconformal mapppings
7. A type criterion
23. Simple examples for value distribution
24. Teichmüller's work on the type problem
1. Introduction: the type problem
2. Line complexes
3. Teichmüller's paper “Eine Anwendung quasikonformer Abbildungen auf das Typenproblem”
4. Nevanlinna's conjecture
5. The type problem in Teichmüller's paper “Untersuchungen über konforme und quasikonforme Abbildungen”
References
25. A Commentary on Teichmüller's paper “Untersuchungen über konformeund quasikonforme Abbildungen”
1. Introduction
2. Module theorems
3. Ahlfors' distortion theorem
4. The Modulsatz
5. The Main Lemma
References
26. Value distribution theory and Teichmüller's paper “Einfache Beispiele zur Wertverteilungslehre”
1. Introduction
2. Value distribution theory before Nevanlinna
3. Nevanlinna's two theorems, and the inverse problem
4. Riemann surfaces
5. On Ahlfors' and Teichmüller's approaches
6. Teichmüller's work
7. Gauss–Lucas and Thurston
References
List of contributors
Index
