Geometric Measure Theory: A Beginner's Guide

✍ Scribed by Frank Morgan

• Publisher: Academic Press Inc
• Year: 2003
• Tongue: English
• Leaves: 226
• Edition: 3 Sub
• Category: Library

✦ Synopsis

Geometric measure theory has become increasingly essential to geometry as well as numerous and varied physical applications. The third edition of this leading text/reference introduces the theory, the framework for the study of crystal growth, clusters of soap bubbles, and similar structures involving minimization of energy. Over the past thirty years, this theory has contributed to major advances in geometry and analysis including, for example, the original proof of the positive mass conjecture in cosmology. This third edition of Geometric Measure Theory: A Beginner's Guide presents, for the first time in print, the proofs of the double bubble and the hexagonal honeycomb conjectures. Four new chapters lead the reader through treatments of the Weaire-Phelan counterexample of Kelvin's conjecture, Almgren's optimal isoperimetric inequality, and immiscible fluids and crystals. The abundant illustrations, examples, exercises, and solutions in this book will enhance its reputation as the most accessible introduction to the subject.

✦ Table of Contents

Cover......Page 1
Table of Contents......Page 6
Preface......Page 8
1. Geometric Measure Theory......Page 12
2. Measures......Page 18
3. Lipschitz Functions and Rectifiable Sets......Page 31
4. Normal and Rectifiable Currents......Page 45
5. The Compactness Theorem and the Existence of Area-Minimizing Surfaces......Page 68
6. Examples of Area-Minimizing Surfaces......Page 76
7. The Approximation Theorem......Page 86
8. Survey of Regularity Results......Page 89
9. Monotonicity and Oriented Tangent Cones......Page 95
10. The Regularity of Area-Minimizing Hypersurfaces......Page 104
11. Flat Chains Modulo
, Varifolds, and (M, ε, υ)-Minimal Sets......Page 112
12. Miscellaneous Useful Results......Page 119
13. Soap Bubble Clusters......Page 127
14. Proof of Double Bubble Conjecture......Page 146
15. The Hexagonal Honeycomb and Kelvin Conjectures......Page 162
16. Immiscible Fluids and Crystals......Page 177
17. Isoperimetric Theorems in General Codimension......Page 184
Solutions to Exercises......Page 188
Bibliography......Page 205
Index of Symbols......Page 218
Name Index......Page 221
Subject Index......Page 223