Geometric Measure Theory, Third Edition: A Beginner's Guide

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. 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 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. "This fourth edition of Geometric Measure Theory: A Beginner's Guide presents the latest results on soap bubble clusters and double bubbles in spheres, tori, and Gauss space. Gauss space, defined as Euclidean space with Gaussian density, of long import to probabilists, appears in Perelman's original paper on the Poincare Conjecture. This edition also describes general manifolds with density and their relationship to Perelman's paper.Throughout there are updates, new illustrations, new exercises and solutions, and new references. Morgan emphasizes geometry over proofs and technicalities to provide the most accessible introduction to the subject."--BOOK JACKET.