Book:L.V. Ahlfors/Conformal Invariants: Topics in Geometric Function Theory
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L.V. Ahlfors: Conformal Invariants: Topics in Geometric Function Theory
Published $\text {1973}$, 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