Theorems on Regularity and Singularity of Energy Minimizing Maps

1 Analytic Preliminaries.- 1.1 Hölder Continuity.- 1.2 Smoothing.- 1.3 Functions with L2 Gradient.- 1.4 Harmonic Functions.- 1.5 Weakly Harmonic Functions.- 1.6 Harmonic Approximation Lemma.- 1.7 Elliptic regularity.- 1.8 A Technical Regularity Lemma.- 2 Regularity Theory for Harmonic Maps.- 2.1 Definition of Energy Minimizing Maps.- 2.2 The Variational Equations.- 2.3 The ?-Regularity Theorem.- 2.4 The Monotonicity Formula.- 2.5 The Density Function.- 2.6 A Lemma of Luckhaus.- 2.7 Corollaries of Luckhaus’ Lemma.- 2.8 Proof of the Reverse Poincaré Inequality.- 2.9 The Compactness Theorem.- 2.10 Corollaries of the ?-Regularity Theorem.- 2.11 Remark on Upper Semicontinuity of the Density ?u(y).- 2.12 Appendix to Chapter 2.- 2.12.1 Absolute Continuity Properties of Functions in W1,2.- 2.12.2 Proof of Luckhaus’ Lemma (Lemma 1 of Section 2.6).- 2.12.3 Nearest point projection.- 2.12.4 Proof of the ?-regularity theorem in case n = 2.- 3 Approximation Properties of the Singular Set.- 3.1 Definition of Tangent Map.- 3.2 Properties of Tangent Maps.- 3.3 Properties of Homogeneous Degree Zero Minimizers.- 3.4 Further Properties of sing u.- 3.5 Definition of Top-dimensional Part of the Singular Set.- 3.6 Homogeneous Degree Zero ? with dim S(?) = n — 3.- 3.7 The Geometric Picture Near Points of sing*u.- 3.8 Consequences of Uniqueness of Tangent Maps.- 3.9 Approximation properties of subsets of ?n.- 3.10 Uniqueness of Tangent maps with isolated singularities.- 3.11 Functionals on vector bundles.- 3.12 The Liapunov-Schmidt Reduction.- 3.13 The ?ojasiewicz Inequality for ?.- 3.14 ?ojasiewicz for the Energy functional on Sn-1.- 3.15 Proof of Theorem 1 of Section 3.10.- 3.16 Appendix to Chapter 3.- 3.16.1 The Liapunov-Schmidt Reduction in a Finite Dimensional Setting.- 4 Rectifiability of the Singular Set.- 4.1 Statement of Main Theorems.- 4.2 A general rectifiability lemma.- 4.3 Gap Measures on Subsets of ?n.- 4.4 Energy Estimates.- 4.5 L2 estimates.- 4.6 The deviation function ?.- 4.7 Proof of Theorems 1, 2 of Section 4.1.- 4.8 The case when ? has arbitrary Riemannian metric.