Geometric Measure Theory (eBook)

A Beginner's Guide

Frank Morgan (Autor)

eBook Download: PDF

2000 | 3. Auflage
227 Seiten
Elsevier Science (Verlag)
978-0-08-052560-0 (ISBN)

Infos

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.

Inhalt

Cover 1
Contents 6
Preface 8
Chapter 1. Geometric Measure Theory 12
Chapter 2. Measures 18
Chapter 3. Lipschitz Functions and Rectifiable Sets 32
Chapter 4. Normal and Rectifiable Currents 46
Chapter 5. The Compactness Theorem and the Existence of Area-Minimizing Surfaces 70
Chapter 6. Examples of Area-Minimizing Surfaces 78
Chapter 7. The Approximation Theorem 88
Chapter 8. Survey of Regularity Results 92
Chapter 9. Monotonicity and Oriented Tangent Cones 98
Chapter 10. The Regularity of Area-Minimizing Hypersurfaces 108
Chapter 11. Flat Chains Modulo v, Varifolds, and (M, e, d)-Minimal Sets 116
Chapter 12. Miscellaneous Useful Results 124
Chapter 13. Soap Bubble Clusters 132
Chapter 14. Proof of Double Bubble Conjecture 152
Chapter 15. The Hexagonal Honeycomb and Kelvin Conjectures 168
Chapter 16. Immiscible Fluids and Crystals 184
Chapter 17. Isoperimetric Theorems in General Codimension 192
Solutions to Exercises 196
Bibliography 214
Index of Symbols 228
Name Index 232
Subject Index 234

Produkt-Details

• Erscheint lt. Verlag: 22.8.2000
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Themenwelt: Wirtschaft
• ISBN-10: 0-08-052560-1 / 0080525601
• ISBN-13: 978-0-08-052560-0 / 9780080525600