Conformal invariants. Topics in geometric function theory

✍ Scribed by Ahlfors L.V.

• Publisher: AMS
• Year: 2010
• Tongue: English
• Leaves: 171
• Series: Ams Chelsea Publishing
• Category: Library

✦ Synopsis

Most conformal invariants can be described in terms of extremal properties. Conformal invariants and extremal problems are therefore intimately linked and form together the central theme of this classic book which is primarily intended for students with approximately a year's background in complex variable theory. The book emphasizes the geometric approach as well as classical and semi-classical results which Lars Ahlfors felt every student of complex analysis should know before embarking on independent research. At the time of the book's original appearance, much of this material had never appeared in book form, particularly the discussion of the theory of extremal length. Schiffer's variational method also receives special attention, and a proof of $\vert a_4\vert \leq 4$ is included which was new at the time of publication. The last two chapters give an introduction to Riemann surfaces, with topological and analytical background supplied to support a proof of the uniformization theorem. Included in this new reprint is a Foreword by Peter Duren, F. W. Gehring, and Brad Osgood, as well as an extensive errata.

✦ Table of Contents

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Contents......Page 4
Foreword......Page 8
Preface......Page 10
1.1. The noneuclidean metric......Page 12
1.2. The Schwarz-Pick theorem......Page 14
1.3. Convex regions......Page 16
1.4. Angular derivatives......Page 18
1.5. Ultra hyperbolic metrics......Page 23
1.6. Bloch's theorem......Page 25
1.8. An elementary lower bound......Page 27
1.9. The Picard theorems......Page 30
2.1. The transfinite diameter......Page 34
2.2. Potentials......Page 35
2.3. Capacity and the transfinite diameter......Page 38
2.4. Subsets of a circle......Page 41
2.5. Symmetrization......Page 42
3.1. The majorization principle......Page 48
3.2. Applications in a half plane......Page 51
3.3. Milloux's problem......Page 52
3.4. The precise form of Hadamard's theorem......Page 55
4.1. Definition of extremal length......Page 61
4.2. Examples......Page 63
4.3. The comparison principle......Page 64
4.4. The composition laws......Page 65
4.5. An integral inequality......Page 67
4.6. Prime ends......Page 68
4.7. Extremal metrics......Page 72
4.8. A case of spherical extremal metric......Page 74
4.9. The explicit formula for extremal distance......Page 76
4.10. Configurations with a single modulus......Page 81
4.11. Extremal annuli......Page 82
4.12. The function A(R)......Page 85
4.13. A distortion theorem......Page 87
4.14. Reduced extremal distance......Page 89
5.1. The area theorem......Page 93
5.2. The Grunsky and Golusin inequalities......Page 96
5.3. Proof of la_41 le 4......Page 98
6.1. Approximation by slit mappings......Page 103
6.3. Proof of la_31 le 3......Page 107
7.1. Variation of the Green's function......Page 109
7.2. Variation of the mapping function......Page 113
7.3. The final theorem......Page 111
7.4. The slit variation......Page 117
8.1. The differential equation......Page 118
8.2. Trajectories......Page 121
8.3. The I' structures......Page 125
8.4. Regularity and global correspondence......Page 127
8.5. The case n = 3......Page 129
9.1. Definition and examples......Page 136
9.2. Covering surfaces......Page 138
9.3. The fundamental group......Page 139
9.4. Subgroups and covering surfaces......Page 141
9.5 Cover transformations......Page 143
9.6 Simply connected surfaces......Page 145
10.1. Existence of the Green's function......Page 147
10.2 Harmonic measure and the maximum principle......Page 149
10.3. Equivalence of the basic conditions......Page 150
10.4 Proof of the uniformization theorem (Part I)......Page 153
10.5. Proof of the uniformization theorem (Part II)......Page 158
10.6. Arbitrary Riemann surfaces......Page 160
Bibliography......Page 163
Index......Page 167
Errata......Page 170