Book:L.V. Ahlfors/Conformal Invariants: Topics in Geometric Function Theory

From ProofWiki
Jump to: navigation, search

L.V. Ahlfors: Conformal Invariants: Topics in Geometric Function Theory

Published $1973$, AMS Chelsea Publishing.


Subject Matter


Contents

Foreword
Preface
1. Applications of Schwarz's lemma
1-1. The noneuclidean metric
1-2. The Schwarz-Pick theorem
1-3. Convex regions
1-4. Angular derivatives
1-5. Ultrahyperbolic metrics
1-6. Bloch's theorem
1-7. The Poincaré metric of a region
1-8. An elementary lower bound
1-9. The Picard theorems
2. Capacity
2-1. The transfinite diameter
2-2. Potentials
2-3. Capacity and the transfinite diameter
2-4. Subsets of a circle
2-5. Symmetrization
3. Harmonic measure
3-1. The majorization principle
3-2. Applications in a half plane
3-3. Milloux's problem
3-4. The precise form of Hadamard's theorem
4. Extremal length
4-1. Definition of extremal length
4-2. Examples
4-3. The comparison principle
4-4. The composition laws
4-5. An integral inequality
4-6. Prime ends
4-7. Extremal metrics
4-8. A case of spherical extremal metric
4-9. The explicit formula for extremal distance
4-10. Configurations with a single modulus
4-11. Extremal annuli
4-12. The function $\Lambda \left({R}\right)$
4-13. A distortion theorem
4-14. Reduced extremal distance
5. Elementary theory of univalent functions
5-1. The area theorem
5-2. The Gunsky and Golusin inequalities
5-3. Proof of $\left\vert{a_4}\right\vert \le 4$
6. Löewner's method
6-1. Approximation by slit mappings
6-2. Löewner's differential equation
6-3. Proof of $\left\vert{a_3}\right\vert \le 3$
7. The Schiffer variation
7-1. Variation of the Green's function
7-2. Variation of the mapping function
7-3. The final theorem
7-4. The slit variation
8. Properties of the extremal functions
8-1. The differential equation
8-2. Trajectories
8-3. The $\Gamma$ structures
8-4. Regularity and global correspondence
8-5. The case $n = 3$
9. Riemann surfaces
9-1. Definition and examples
9-2. Covering surfaces
9-3. The fundamental group
9-4. Subgroups and covering surfaces
9-5. Cover transformations
9-6. Simply connected surfaces
10. The uniformization theorem
10-1. Existence of the Green's function
10-2. Harmonic measure and the maximum principle
10-3. Equivalence of the basic conditions
10-4. Proof of the uniformization theorem (Part I)
10-5. Proof of the uniformization theorem (Part II)
10-6. Arbitrary Riemann surfaces
Bibliography
Index
Errata