Lectures on Geometric Variational Problems

ISBN-13: 9784431701521 / Angielski / Miękka / 1996 / 154 str.

Seiki Nishikawa;Richard Schoen

Opis

In this volume are collected notes of lectures delivered at the First In ternational Research Institute of the Mathematical Society of Japan. This conference, held at Tohoku University in July 1993, was devoted to geometry and global analysis. Subsequent to the conference, in answer to popular de mand from the participants, it was decided to publish the notes of the survey lectures. Written by the lecturers themselves, all experts in their respective fields, these notes are here presented in a single volume. It is hoped that they will provide a vivid account of the current research, from the introduc tory level up to and including the most recent results, and will indicate the direction to be taken by future researeh. This compilation begins with Jean-Pierre Bourguignon's notes entitled "An Introduction to Geometric Variational Problems," illustrating the gen eral framework of the field with many examples and providing the reader with a broad view of the current research. Following this, Kenji Fukaya's notes on "Geometry of Gauge Fields" are concerned with gauge theory and its applications to low-dimensional topology, without delving too deeply into technical detail. Special emphasis is placed on explaining the ideas of infi nite dimensional geometry that, in the literature, are often hidden behind rigorous formulations or technical arguments."

Spis treści

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.