Lectures on Geometric Variational Problems -

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Résumé :
The field of geometric variational problems, that is, nonlinear problems arising in geometry and topology from the point of view of global analysis, has developed very rapidly in the last decade. It was therefore felt timely to produce a set of presentations on this subject in which leading experts would provide general surveys of current research from the fundamentals to the most recent results with a view to future research. This volume will interest both mature researchers and graduate students concerned with gauge theory and low dimensional topology, theory of harmonic maps, minimal surfaces, and minimal submanifolds in Riemannian manifolds.

Sommaire:
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.