Ahlfors / Kra / Maskit Contributions to Analysis

A Collection of Papers Dedicated to Lipman Bers
1. Auflage 2014
ISBN: 978-1-4832-6116-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 460 Seiten, Web PDF

Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers is a compendium of papers provided by Bers, friends, students, colleagues, and professors. These papers deal with Teichmuller spaces, Kleinian groups, theta functions, algebraic geometry. Other papers discuss quasiconformal mappings, function theory, differential equations, and differential topology. One paper discusses the results of the rigidity theorem of Mostow and its generalization by Marden in relation to geometric properties of Kleinian groups of the first kind. These results, obtained by planar methods, are presented in terms of the hyperbolic 3-space language, which is a natural pedestal in approaching the action of the Kleinian groups. Another paper reviews Riemann's vanishing theorem which solves the Jacobi inversion problem, by relating the vanishing properties of the theta function (particularly at half periods) to properties of certain linear series on the Riemann surface. One paper examines the problem of obtaining relations among the periods of the differentials of first kind on a compact Riemann surface. An application of a computer program involves supersonic transport. The program is based on the hodograph transformation and a method of complex characteristics to calculate profiles that are shock-less at a specified angle of attack, or at a specified subsonic free-stream Mach number. The collection can prove useful for engineers, statisticians, students, and professors in advance mathematics or courses related to aeronautics.

Autoren/Hrsg.

Ahlfors, Lars V.

Kra, Irwin

Maskit, Bernard

Inhaltsverzeichnis

1;Front Cover;1
2;Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers;4
3;Copyright Page;5
4;Table of Contents;8
5;Contributors;3
6;List of Contributors;16
7;PREFACE;18
8;Chapter 1. On the Decomposition and Deformation of Kleinian Groups;20
8.1;Introduction;20
8.2;1. Some Kleinian Groups;22
8.3;2. The Maskit Combination Theorems;23
8.4;3. The Decomposition of Finitely Generated Kleinian Groups of the Second Kind;24
8.5;4. Deformations of Finitely Generated Kleinian Groups;25
8.6;References;28
9;Chapter 2. Some Loci of Teichmuller Space for Genus Five Defined by
Vanishing Theta Nulls;30
9.1;1. Introduction;30
9.2;2. Riemann's Vanishing Theorem;31
9.3;3. g13s
on Surfaces of Genus Five;32
9.4;4. Surfaces of Genus
Five Admitting Two Half-Canonical g41s;33
9.5;5. Proofs of the
Theorems;34
9.6;6. Necessary Conditions for Elliptic-Hyperellipticity;35
9.7;7. Some Remarks on the "p — 2 Conjecture" for Higher Genus;36
9.8;References;37
10;Chapter 3. Conditions for Quasiconformal Deformations in Several Variables;38
10.1;1. Introduction;38
10.2;2. Uniqueness;39
10.3;3. The Adjoint Operator;40
10.4;4. Statement of the Theorem;41
10.5;5. Necessity;42
10.6;6. Sufficiency;43
10.7;References;44
11;Chapter 4. On Locally Quasiconformal Mappings in Space (n
= 3);46
11.1;References;49
12;Chapter 5. A Theorem on the Boundary Correspondence under Conformal
Mapping with Application to Free Boundary Problems of Fluid Dynamics;50
12.1;Introduction;50
12.2;1. Boundary Correspondence Theorem;51
12.3;2. Application to Cavity Flows;54
12.4;References;57
13;Chapter 6. The Extension Problem for Quasiconformal Mappings;58
13.1;References;66
14;Chapter 7. Hyperbolic Spaces;68
14.1;1. Introduction and Preliminaries;68
14.2;2. The Hyperbolic Space
Hn(F);71
14.3;3. Conjugacy Classes in
U(1, n; F);81
14.4;4. Subgroups ofc U(1, n;
F);92
14.5;References;105
15;Chapter 8. On .-Monogenic Functions, and the Mean Value
Theorem of the Differential Calculus;108
15.1;1. Reminiscence, and
.-Monogenic Functions;108
15.2;2. Remarks on the Elementary Mean Value Theorem of the Differential Calculus;111
15.3;Abstract (of Lecture);112
15.4;Lecture;112
15.5;References;117
16;Chapter 9. On Quasiconformal Extensions of the Beurling-Ahlfors Type;118
16.1;1. Introduction;118
16.2;2. Statement of Theorems;118
16.3;3. Proof of Theorem 1;120
16.4;4. Proof of Theorem 2;122
16.5;5. Proof of Theorem 3;123
16.6;References;124
17;Chapter 10. On Holomorphic Mappings between Teichmüller Spaces;126
17.1;Introduction;126
17.2;1. Teichmüller Spaces;126
17.3;2. The Teichmüller Metric;129
17.4;3. The Automorphism Group of T(g,n);132
17.5;4. The Teichmüller Curve and Its Sections;135
17.6;5. Most Fiber Spaces Are Not Teichmüller Spaces;141
17.7;References;143
18;Chapter 11. On the Poincare Relation;144
18.1;Introduction;144
18.2;References;151
19;Chapter 12. Elliptic Functions and Modular Forms;152
19.1;Introduction;152
19.2;Reference;164
20;Chapter 13. On the Differentiability of Solutions of Accretive Linear Differential Equations;166
20.1;References;169
21;Chapter 14. Survey of Some Recent Progress in Transonic Aerodynamics;170
21.1;1. Introduction;170
21.2;2. Optimal Transonic Airfoils;171
21.3;3. Three-Dimensional Problems;175
21.4;References;177
22;Chapter 15. The Hausdorff Measure of Sets Which Link in Euclidean Space;178
22.1;1. Introduction;178
22.2;2. Preliminary Results;179
22.3;3. Main Construction;181
22.4;4. Main Results;184
22.5;5. An Inequality for Quasiconformal Mappings;185
22.6;References;186
23;Chapter 16. Two Results in the Global Theory of Holomorphic Mappings;188
23.1;Introduction;188
23.2;1. Statement of Theorem I;189
23.3;2. The First Main Theorem; A Result of Chern-Wu;190
23.4;3. Two Estimates from the Theory of Holomorphic Curves;192
23.5;4. Proof of Theorem I;196
23.6;5. Some Comments and Examples;197
23.7;6. Statement and Proof of Theorem II;199
23.8;7. Some Comments and Questions regarding Holomorphic Curves;200
23.9;References;202
24;Chapter 17. On Fundamental Domains and the Teichmüller Modular Group;204
24.1;1. Introduction;204
24.2;2. The Torus with a Hole;205
24.3;3. Geodesies on S;208
24.4;4. The Twist Parameter;209
24.5;References;212
25;Chapter 18. Differential Equations in a Projective Space and Linear Dependence over a Projective Variety;214
25.1;Introduction;214
25.2;1. Canonical Characteristic Sets;215
25.3;2. Differentially Homogeneous and Differentially Multihomogeneous Differential Polynomials;216
25.4;3. Differentially Homogeneous and Differentially Multihomogeneous Differential Ideals;217
25.5;4. Algebraic Differential Equations in Projective and Multiprojective Spaces;222
25.6;5. Differential Specializations;223
25.7;6. Differentially Complete Differentially Closed Sets;225
25.8;7. Linear Dependence over Projective Varieties;229
25.9;8. Examples;232
25.10;References;233
26;Chapter 19. On the Complexes on the Boundary Induced by Elliptic Complexes of Differential Operators;234
26.1;Introduction;234
26.2;0. Preliminaries;235
26.3;1. Normal Derivative Operators;245
26.4;2. Operators Induced on the Boundary;250
26.5;3. Operators on the Boundary Induced by the Elliptic Complex of Differential Operators;254
26.6;References;259
27;Chapter 20. Group Isomorphisms Induced by Quasiconformal Mappings;260
27.1;References;263
28;Chapter 21. Partial Differential Equations Invariant under Conformai or Projective Transformations;264
28.1;Introduction;264
28.2;Part I;267
28.2.1;1. Derivation of Conditions (4) and (5) for Partial Conformai Invariance of (3);267
28.2.2;2. The Dirichlet Problem for (6') in Domains with Smooth Compact Boundaries;268
28.2.3;3. A Priori Estimates for Nonnegative Solutions of (2.1);270
28.2.4;4. Solutions of (6') Which Are Infinite on the Boundary;272
28.2.5;5. Solutions of (6') for Arbitrary Domains in Rn;274
28.3;Part II;278
28.3.1;6. Invariance of (9) and (10) under Projective Transformations;278
28.3.2;7. The Boundary Value Problem (9);281
28.3.3;8. A Priori Estimates for Second Derivatives;283
28.3.4;9. The Metric (10) Is Complete;288
28.3.5;10. The Mean Curvature of the Surface Is Bounded;289
28.4;References;291
29;Chapter 22. Schottky Groups and Circles;292
29.1;1. Introduction;292
29.2;2. Schottky Space;293
29.3;3. Classical Schottky Space;294
29.4;4. Convergence of Classical Schottky Groups;294
29.5;5. Proof of the Proposition;295
29.6;6. Proof of Theorem 3.1;297
29.7;7. Fuchsian Groups of the Second Kind;297
29.8;References;297
30;Chapter 23. Homomorphisms Associated with Multiplicative Functions;298
30.1;Introduction;298
30.2;1. Complex Tori and Multiplicative Functions;299
30.3;2. a(.,
F);305
30.4;References;310
31;Chapter 24. Uniformizations of Riemann Surfaces;312
31.1;Introduction;312
31.2;1. What Is a Uniformization?;313
31.3;2. Isomorphisms;315
31.4;3. Factor Subgroups;315
31.5;4. Precisely Invariant Sets;317
31.6;5. Picture of a Uniformization;318
31.7;6. Standard Uniformizations;320
31.8;7. Torsion-Free Uniformizations;325
31.9;8. The General Uniformization Theorem;327
31.10;References;331
32;Chapter 25. Some Restrictions on the Smooth Immersion of Complete Surfaces in E3;332
32.1;Introduction;332
32.2;1. Definitions;333
32.3;2. Preliminary Results;336
32.4;3. Some Theorems;339
32.5;4. Subsidiary Results;341
32.6;References;342
33;Chapter 26. Prym Varieties I;344
33.1;Introduction;344
33.2;Notations;345
33.3;Part I;345
33.3.1;1. Double Coverings of Curves;345
33.3.2;2. A Configuration of Abelian Varieties;347
33.3.3;3. Definition of the Prym Variety;350
33.4;Part II;352
33.4.1;4. Relations between Theta Divisors;352
33.4.2;5. The Splitting of
.-1(.Y,y)for Jacobians;356
33.5;Part III;360
33.5.1;6. Geometric Description of Sing
Unramified Case;360
33.5.2;7. Dim
Sing;363
33.6;Appendix : A Theorem of Martens;367
33.7;References;369
34;Chapter 27. Asymptotic Decay for Ultrahyperbolic Operators;370
34.1;1. Introduction;370
34.2;2. The Ultrahyperbolic Operator;371
34.3;3. Generalized Ultrahyperbolic Operators;373
34.4;References;374
35;Chapter 28. Instability of Thin-Walled Spherical Structures under External Pressure;376
35.1;1. Introduction;376
35.2;2. Basic Equations;378
35.3;3. Method of Solution of the Equations;380
35.4;4. A Numerical Example;384
35.5;5. Legendre Functions and the Coefficients (l,
m, n);387
35.6;References;391
36;Chapter 29. Extremal
Quasiconformal Mappings with Given Boundary Values;394
36.1;Introduction;394
36.2;1. Quasiconformal Mappings of the Unit Disc with the Same Boundary Values;395
36.3;2. The Sufficiency of Hamilton's Condition;399
36.4;3. Proof of Hamilton's Theorem (Necessity for Extremality);402
36.5;4. Some Consequences of Hamilton's Condition;406
36.6;References;410
37;Chapter 30. Invariant Metrics on Teichmüller Space;412
37.1;1. Differential Metrics on Teichmüller Space;412
37.2;2. Hermitian Metrics on Tg;413
37.3;3. Metrics Associated with the Embedding of Tg in the Siegel Upper Half-Plane;415
37.4;References;418
38;Chapter 31. A Constructive Proof of the Riemann-Roch Theorem for Curves;420
38.1;References;424
39;Chapter 32. Function Theory on Differentiable Submanifolds;426
39.1;1. Introduction;426
39.2;2. Holomorphic Approximation and Extension: Examples;427
39.3;3. Holomorphic Approximation;432
39.4;4. Envelopes of Holomorphy and Holomorphic Extension;441
39.5;5. CR Function Theory;449
39.6;References;456