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

Subject Matter

 * Functional Analysis

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