Geometric Measure Theory: A Beginner's Guide
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Table of contents
- Cover
- Contentsv
- Prefacevii
- Chapter 1. Geometric Measure Theory1
- Chapter 2. Measures7
- Chapter 3. Lipschitz Functions and Rectifiable Sets21
- Chapter 4. Normal and Rectifiable Currents35
- Chapter 5. The Compactness Theorem and the Existence of Area-Minimizing Surfaces59
- Chapter 6. Examples of Area-Minimizing Surfaces67
- Chapter 7. The Approximation Theorem77
- Chapter 8. Survey of Regularity Results81
- Chapter 9. Monotonicity and Oriented Tangent Cones87
- Chapter 10. The Regularity of Area-Minimizing Hypersurfaces97
- Chapter 11. Flat Chains Modulo v, Varifolds, and (M, ε, δ)-Minimal Sets105
- Chapter 12. Miscellaneous Useful Results113
- Chapter 13. Soap Bubble Clusters121
- Chapter 14. Proof of Double Bubble Conjecture141
- Chapter 15. The Hexagonal Honeycomb and Kelvin Conjectures157
- Chapter 16. Immiscible Fluids and Crystals173
- Chapter 17. Isoperimetric Theorems in General Codimension181
- Solutions to Exercises185
- Bibliography203
- Index of Symbols217
- Name Index221
- Subject Index223
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.
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.