Lectures on Geometric Variational Problems

An Introduction to Geometric Variational Problems).- I The General Setting.- 1. A General Framework.- 2. A Rough Classification of Geometric Variational Problems.- 3. Different Points of View on Geometric Variational Problems and Their Uses.- II A Review of Geometric Variational Problems.- 1. The Eigenvalue Problem for the Laplacian on Functions.- 2. Harmonic Forms.- 3. Length and Energy of Curves.- 4. The Energy of Maps.- 5. Minimal Submanifolds.- 6. Yang-Mills Fields.- 7. The Total Scalar Curvature Functional.- 8. Some Other Non-Local Functionals of Riemannian Metrics.- III Symmetry Considerations, Topological Constraints, and Interactions with Physics.- 1. Symmetry Considerations.- 2. Topological Constraints.- 3. Interactions with Physics.- Geometry of Gauge Fields.- §1 Donaldson Invariant of 4 Manifolds.- §2 Basic Properties of the Moduli Space of ASD Connections.- §3 Casson Invariant and Gauge Theory.- §4 Floer Homology.- §5 Donaldson Invariant as Topological Field Theory.- §6 Gauge Theory on 4 Manifolds with Product End.- §7 Equivariant Floer Theory, Higher Boundary and Degeneration at Infinity.- Theorems on the Regularity and Singularity of Minimal Surfaces and Harmonic Maps.- LECTURE 1 Basic Definitions, and the ?-Regularity and Compactness Theorems.- 1 Basic Definitions and the ?-Regularity and Compactness Theorems.- 2 The ?-Regularity and Compactness Theorems.- LECTURE 2 Tangent Maps and Affine Approximation of Subsets of Rn.- 1 Tangent Maps.- 2 Properties of Homogeneous Degree Zero Minimizers.- 3 Approximation of Subsets of Rn by Affine Subspaces.- 4 Further Properties of sing u.- 5 Homogeneous Degree Zero ? with dim S(?) = n — 3.- LECTURE 3 Asymptotics on Approach to Singular Points.- 1 Significance of Unique Asymptotic Limits.- 2 Lojasiewicz Inequalities for the Energy.- 3 Proof of Theorem 1 of §1.- LECTURE 4 Recent Results on Rectifiability and Smoothness Properties of sing u.- 1 Statement of Main Theorems.- 2 Brief Discussion of Techniques.