Geometric measure theory: a beginner’s guide

Frank Morgan9780080525600, 9780125068512, 0125068514

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.

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Table of contents :
Cover ……Page 1
Date-line ……Page 2
Contents ……Page 3
Preface ……Page 4
1. Geometric Measure Theory ……Page 7
2. Measures ……Page 13
3. Lipschitz Functions and Rectifiable Sets ……Page 27
4. Normal and Rectifiable Currents ……Page 41
5. The Compactness Theorem and the Existence of Area-Minimizing Surfaces ……Page 65
6. Examples of Area-Minimizing Surfaces ……Page 73
7. The Approximation Theorem ……Page 83
8. Survey of Regularity Results ……Page 87
9. Monotonicity and Oriented Tangent Cones ……Page 93
10. The Regularity of Area-Minimizing Hypersurfaces ……Page 103
11. Flat Chains Modulo $nu$, Varifolds, and $(M,varepsilon,delta$-Minimal Sets ……Page 111
12. Miscellaneous Useful Results ……Page 119
13. Soap Bubble Clusters ……Page 127
14. Proof of Double Bubble Conjecture ……Page 147
15. The Hexagonal Honeycomb and Kelvin Conjectures ……Page 163
16. Immiscible Fluids and Crystals ……Page 179
17. Isoperimetric Theorems in General Codimension ……Page 187
Solutions to Exercises ……Page 191
Bibliography ……Page 209
Index of Symbols ……Page 223
Name Index ……Page 227
Subject Index ……Page 229