نگاشت‌های کوآسی‌فرمال، سطوح ریمان و فضاهای تیشمولر: جلسه ویژه AMS به افتخار کلیفورد جی. ارل، ۲-۳ اکتبر ۲۰۱۰، دانشگاه سیراکوز، نیویورک

This volume contains the proceedings of the AMS Special Session on Quasiconformal Mappings, Riemann Surfaces, and Teichmüller Spaces, held in honor of Clifford J. Earle, from October 2–3, 2010, in Syracuse, New York. This volume includes a wide range of papers on Teichmüller theory and related areas. It provides a broad survey of the present state of research and the applications of quasiconformal mappings, Riemann surfaces, complex dynamical systems, Teichmüller theory, and geometric function theory. The papers in this volume reflect the directions of research in different aspects of these fields and also give the reader an idea of how Teichmüller theory intersects with other areas of mathematics. Preface 8 Some remarks on singly degenerate Kleinian groups 10 1. Thurston changes the nature of the playing field 11 2. Preliminaries 11 3. Fuchsian Groups and Collapsed Laminations 12 4. Torsion-free singly degenerate groups and the quotient 3-manifold 14 5. Known results about the limit sets of singly degenerate groups 15 6. Local connectivity and its consequenses 15 7. Bringing it all together in a picture 18 8. The Sullivan Dictionary and local connnectedness 19 References 20 On a theorem of Kas and Schlessinger 22 1. Preliminaries 23 2. The theorem of A. Kas and M. Schlessinger 27 3. Flat families of nodal curves 30 References 30 Conformally scattered sets in the unit circle 32 1. Introduction 32 2. The Cantor-Bendixson derivative and conformally scattered sets 33 3. Conformal towers in the SL(2,Z)-orbit of ∞ 35 4. Appendix: Ordinal Numbers 37 References 38 Finiteness conditions on translation surfaces 40 Introduction 40 1. Inequivalence of finiteness conditions 41 2. Discreteness of Veech groups 44 References 48 Holomorphic plumbing coordinates 50 1. Review of the space of tori 50 2. Classical holomorphic plumbing 51 3. Teichmüller space T(R) 51 4. Noded surfaces and their Teichmüller spaces 52 5. Pinch maps 52 6. Augmented Teichmüller space and its horocyclic topology 53 7. Boundary spaces 54 8. The Bers slice 55 9. Opening the nodes 55 10. Part 1: Choosing felicitous local coordinate charts 55 11. The quotient manifold 56 12. Part 2: Analytic continuation of plumbing vector 57 13. The main theorems 58 14. Applications 59 References 60 On Böttcher coordinates and quasiregular maps 62 1. Introduction 62 2. Statement of results 63 3. Preliminaries 64 4. Proof of Theorem 2.1 68 5. Logarithmic transforms of ψ_{k} 69 6. Proof of Theorem 2.4 79 7. Proof of Theorem 2.7 80 References 84 Discontinuity of asymptotic Teichmüller modular group 86 1. Introduction 86 2. Preliminaries 87 3. Projection of the limit set 90 4. Projection of the region of discontinuity 91 5. Non-empty region of discontinuity 93 References 95 Extremal annuli on the sphere 98 Introduction 98 1. The intersection inequality 101 2. The minimal axis theorem 102 3. The Teichmüller annulus 104 4. The Mori annulus 108 5. Comparison of the Mori and Teichmüller annuli 112 6. Pairs of extremal annuli on a four times punctured sphere 114 7. Mori type extremal problems 114 References 116 Lifting free subgroups of PSL(2,R) to free groups 118 1. Introduction 118 2. F-sequences 120 3. Word forms and primitive exponents 122 4. Two generator subgroups of PSL(2,R) 124 5. Primitive exponents 127 6. Lifting to the free group 129 7. The Geometries of Different Representations 129 References 130 An introduction to Beauville surfaces via uniformization 132 1. Introduction 132 2. Triangle groups and triangle G-coverings 133 3. The concept of Beauville surface 142 4. Uniformization of Beauville surfaces: unmixed case 143 5. Uniformization of Beauville surfaces: mixed case 150 6. Unmixed Beauville surfaces with group PSL(2,p) and bitype ((2,3,n),(p,p,p)) 157 References 159 Symmetry and moduli spaces for Riemann surfaces 162 1. Introduction 162 2. Background algebraic geometry 164 3. The Deligne-Mumford compactification 165 4. The moduli spaces of pointed spheres M_{0,n} 168 5. A genus 2 modular subvariety 175 6. Final Comments on modular symmetry 176 References 177 Conformally natural extensions of continuous circle maps: I. The case when the pushforward measure has no atom 180 1. Introduction 180 2. Conformal barycenter 183 3. Conformally barycentric extension 186 4. Partial derivatives and related 189 5. Proofs of Theorems 2 and 3 192 References 207 Normal and quasinormal families of quasiregular mappings 208 1. Introduction 208 2. Preliminaries 210 3. Proofs Of The Theorems 212 References 218 Symmetric invariant measures 220 1. Uniformly symmetric circle endomorphisms 220 2. Symmetric invariant measures 221 3. Dini smoothness and expanding Blaschke products 223 References 227 Douady-Earle section, holomorphic motions, and some applications 228 Introduction 228 Part 1. Applications of Douady-Earle section in holomorphic motions 229 1. Basic definitions 229 2. Teichmüller space of a closed set in ̂\C 231 3. Douady-Earle section 233 4. Universal holomorphic motion of a closed set in ̂\C 236 5. Extensions of holomorphic motions 237 6. Extending holomorphic motions and lifting holomorphic maps 243 Part 2. Some applications of holomorphic motions in complex analysis 245 7. An application of Theorem 4.2 245 8. Gluing germs in the Riemann sphere 245 ऀ⸀ 䬀渀椀朠ᤀ猀 琀栀攀漀爀攀洀Ⰰ 䈀琀琀挀栀攀爠ᤀ猀 琀栀攀漀爀攀洀Ⰰ 愀渀搀 琀栀攀椀爀 最攀渀攀爀愀氀椀稀愀琀椀漀渀 248 10. Leau-Fatou flowers and linearization 252 11. Quasiconformal rigidity for parabolic germs 255 References 259 Cook-hats and crowns 262 1. Introduction 262 2. Cook-hats 264 3. Crowns 269 References 271 On cohomology of Kleinian groups V: b-groups 272 1. Introduction and background 272 2. Summary of main results 276 3. Parabolic invariants and surfaces represented by a b-group 277 4. The deficiency δ(q) 279 5. Differentiation: D^{2q-1} 279 6. Global holomorphic Eichler integrals 280 7. Bers’s L-operator and function groups 283 ࠀ⸀ 匀琀爀甀挀琀甀爀攀 漀昀 ⠃꤀ⴃ鐠耀⤠稀丿 286 ऀ⸀ 吀栀攀 昀漀甀爀 挀氀愀猀猀攀猀 漀昀⃘㗜伀ⴀ最爀漀甀瀀 288 References 289 Fundamental inequalities of Reich-Strebel and triangles in a Teichmüller space 292 1. Introduction 292 2. Preliminaries and Notation 293 3. New Versions of the Fundamental Inequalities 295 4. Strong Triangle Inequalities 297 5. Angles Between Two Geodesic Rays 302 References 305 The Petersson series vanishes at infinity 308 1. Introduction 308 2. Basic estimate 310 3. Comparison of euclidean areas 313 4. Proof of the main theorem 316 5. Application to the variation of length functions 318 6. Remarks on vanishing at infinity 319 References 320 On fiber spaces over Teichmüller spaces 322 Introduction 322 1. Basic definitions and results 322 2. Torsion free case 326 3. Torsion case 331 References 336 On the number of holomorphic families of Riemann surfaces 340 1. Introduction 340 2. Preliminaries and statements of main results 341 3. Proof of Theorem 2.2 343 4. Proof of Theorem 2.3 348 References 351 Veech groups of flat structures on Riemann surfaces 352 1. Introduction 352 2. Definitions 353 3. Examples of Veech groups 354 4. Veech groups of coverings of P_{2n} and Universal Veech group of P_{2n} 355 5. Calculation of Veech groups 359 6. Calculation of H/Γ(X) 363 7. Veech groups of Abelian coverings 365 8. Examples 366 References 371 On families of holomorphic differentials on degenerating annuli 372 1. Introduction 372 2. The analytic geometry of zw=t 373 3. Sections of powers of the relative dualizing sheaf; families of regular k-differentials 376 References 379 Transformations of spheres without the injectivity assumption 380 1. Introduction 380 2. Proof of Theorem 1 381 References 384 Beginning as a specialized branch of geometric function theory, Teichmüller theory has become an important field of research in such areas of mathematics as topology, geometry, dynamics, and hyperbolic manifolds; it has even turned up in physics. The 24 papers in this collection include surveys and research reports. Among the topics are conformally scattered sets in the unit circle, extreme annuli on the sphere, normal and quasinormal families of quasiregular mappings, fundamental inequalities of Reich-Strebel and triangles in a Teichmüller space, and the number of holomorphic families of Riemann surfaces. No index is provided. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com)