Preface -- Contents -- Chapter 1 Introduction -- 1.1 Harmonic Maps -- 1.2 Related Elliptic Equations -- 1.3 Evolution Problems -- 1.4 Notation -- Chapter 2 Analytic Preliminaries -- 2.1 A Few Basic Inequalities Involving Sobolev Spaces -- 2.2 The Hodge Decomposition -- 2.3 Bounded Mean Oscillation -- 2.4 The Hardy Space H1(Rm) -- 2.5 A Gagliardo-Nirenberg Type Inequality -- 2.6 A Gehring Type Lemma -- Chapter 3 Harmonic Maps -- 3.1 The Nearest Point Projection and the Second Fundamental Form -- 3.2 The Euler-Lagrange Equation -- 3.3 Stationary Weakly Harmonic Maps and the Monotonicity Formula -- 3.4 Higher Regularity -- 3.5 The Target Manifold Sn-1 -- 3.6 A Moving Tangent Frame Field -- 3.7 General Target Manifolds -- 3.8 Generalized Harmonic Maps -- 3.9 Further Remarks -- 3.9.1 Homogeneous Manifolds -- 3.9.2 Energy Minimizing Maps -- Chapter 4 Almost Harmonic Maps -- 4.1 The Monotonicity Formula -- 4.2 Hölder Continuity -- 4.3 A Lemma of S. Luckhaus -- 4.4 More About the Tangent Frame Field -- 4.5 An Energy Inequality -- 4.6 Small Dimensions -- Chapter 5 Evolution Problems -- 5.1 The Stability Condition for the Harmonic Map Heat Flow -- 5.2 Higher Regularity for Evolution Problems -- 5.3 The Heat Flow into Spheres -- 5.4 The Heat Flow into General Target Manifolds -- 5.5 The Landau-Lifshitz Equation -- 5.6 Further Remarks -- 5.6.1 The Harmonic Map Heat Flow into Homogeneous Manifolds -- 5.6.2 The Landau-Lifshitz Equation for Almost-Kähler Manifolds -- Bibliography -- Index.