Calculus of Variations and Harmonic Maps

Description

Title page Preface to the English Edition Preface Chapter 1. Calculus of Variations 1. The aims of this book 2. Methods of variations and field theories 3. Examples of the method of variations 4. A guide to the further study of the calculus of variations Exercises Classical mechanics Chapter 2. Manifolds 1. Continuity, differentiation, and integration 2. C^k-manifolds 3. Finite-dimensional C^infty-manifolds 4. Examples of manifolds Exercises Chapter 3. Morse Theory 1. Critical points of a smooth function 2. Minimum values of smooth functions 3. The condition (C) 4. An application to closed geodesics The isoperimetric problem and Oueen Dido Chapter 4. Harmonic Mappings 1. What is a harmonic mapping? 2. An alternative expression for the first variation 3. Examples of harmonic mappings Exercises < Coffee Break . Soap films and minimal surfaces (Plateau's problem) Chapter 5. The Second Variation Formula and Stability 1. The second variation formula 2. Instability theorems 3. Stability of holomorphic mappings Exercises Chapter 6. Existence, Construction, and Classification of Harmonic Maps 1. Existence, construction, and classification problems 2. The case of the unit sphere 3. The case of symmetric spaces 4. Proof of the Eells-Sampson theorem via the variational method Solutions to Exercises References Subject Index