Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- 1 Preliminaries -- 1.1. Harmonic Mappings -- 1.2. Some Basic Facts -- 1.3. The Argument Principle -- 1.4. The Dirichlet Problem -- 1.5. Conformal Mappings -- 1.6. Overview of Harmonic Mapping Theory -- 2 General Properties of Harmonic Mappings -- 2.1. Critical Points of Harmonic Functions -- 2.2. Lewy's Theorem -- 2.3. Heinz's Lemma -- 2.4. Rado's ́Theorem -- 2.5. Counterexamples in Higher Dimensions -- 2.6. Approximation Theorem -- 3 Harmonic Mappings onto Convex Regions -- 3.1. The Radó-Kneser-Choquet Theorem -- 3.2. Choquet's Proof -- 3.3. Boundary Behavior -- 3.4. The Shear Construction -- 3.5. Structure of Convex Mappings -- 3.6. Covering Theorems and Coefficient Bounds -- 3.7. Failure of the Radó-Kneser-Choquet Theorem in R -- 4 Harmonic Self-Mappings of the Disk -- 4.1. Representation by Radó-Kneser-Choquet Theorem -- 4.2. Mappings onto Regular Polygons -- 4.3. Arbitrary Convex Polygons -- 4.4. Sharp Form of Heinz's Inequality -- 4.5. Coefficient Estimates -- 4.6. Schwarz's Lemma for Harmonic Mappings -- 5 Harmonic Univalent Functions -- 5.1. Normalizations -- 5.2. Normal Families -- 5.3. The Harmonic Koebe Function -- 5.4. Coefficient Conjectures -- 6 Extremal Problems -- 6.1. Minimum Area -- 6.2. Covering Theorems -- 6.3. Estimation of 2

8.5. Integral Means -- 9 Minimal Surfaces -- 9.1. Background in Surface Theory -- 9.2. Isothermal Parameters -- 9.3. Weierstrass-Enneper Representation -- 9.4. Some Examples -- 9.5. Historical Notes -- 10 Curvature of Minimal Surfaces -- 10.1. Gauss Curvature -- 10.2. Minimal Graphs and Harmonic Mappings -- 10.3. Heinz's Lemma and Bounds on Curvature -- 10.4. Sharp Bounds on Curvature -- 10.5. Schwarzian Derivatives -- Appendix Extremal Length -- References -- Index.