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

Geometric measure theory provides the framework to understand the structure of a crystal, a soap bubble cluster, or a universe. Measure Theory: A Beginner's Guide is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Morgan emphasizes geometry over proofs and technicalities providing a fast and efficient insight into many aspects of the subject. New to the 4th edition:* Abundant illustrations, examples, exercises, and solutions.* The latest results on soap bubble clusters, including a new chapter on "Double Bubbles in Spheres, Gauss Space, and Tori."* A new chapter on "Manifolds with Density and Perelman's Proof of the Poincar? Conjecture."* Contributions by undergraduates. Cover Page......Page 1 Title Page......Page 4 Copyright Page......Page 5 Preface......Page 8 Contents......Page 6 1 Geometric Measure Theory......Page 10 2 Measures......Page 18 3 Lipschitz Functions and Rectifiable Sets......Page 32 4 Normal and Rectifiable Currents......Page 46 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 90 9 Monotonicity and Oriented Tangent Cones......Page 96 10 The Regularity of Area-Minimizing Hypersurfaces......Page 104 11 Flat Chains Modulo ν, Varifolds, and (M, ε, δ)-Minimal Sets......Page 112 12 Miscellaneous Useful Results......Page 118 13 Soap Bubble Clusters......Page 126 14 Proof of Double Bubble Conjecture......Page 148 15 The Hexagonal Honeycomb and Kelvin Conjectures......Page 164 16 Immiscible Fluids and Crystals......Page 178 17 Isoperimetric Theorems in General Codimension......Page 184 18 Manifolds with Density and Perelman’s Proof of the Poincaré Conjecture......Page 188 19 Double Bubbles in Spheres, Gauss Space, and Tori......Page 202 Solutions to Exercises......Page 210 Bibliography......Page 232 Index of Symbols......Page 250 Name Index......Page 252 Subject Index......Page 254 Geometric Measure Theory, Fourth Edition, is an excellent text for introducing ideas from geometric measure theory and the calculus of variations to beginning graduate students and researchers.This updated edition contains abundant illustrations, examples, exercises, and solutions; and the latest results on soap bubble clusters, including a new chapter on Double Bubbles in Spheres, Gauss Space, and Tori. It also includes a new chapter on Manifolds with Density and Perelman's Proof of the Poincaré Conjecture.This text is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Morgan emphasizes geometry over proofs and technicalities providing a fast and efficient insight into many aspects of the subject.New to the 4th edition:• Abundant illustrations, examples, exercises, and solutions.• The latest results on soap bubble clusters, including a new chapter on'Double Bubbles in Spheres, Gauss Space, and Tori.'• A new chapter on'Manifolds with Density and Perelman's Proof of the Poincaré Conjecture.'• Contributions by undergraduates. Geometric measure theory provides the framework to understand the structure of a crystal, a soap bubble cluster, or a universe. Measure Theory: A Beginner's Guide is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Morgan emphasizes geometry over proofs and technicalities providing a fast and efficient insight into many aspects of the subject.

New to the 4th edition:
* Abundant illustrations, examples, exercises, and solutions.
* The latest results on soap bubble clusters,
including a new chapter on "Double Bubbles in
Spheres, Gauss Space, and Tori."
* A new chapter on "Manifolds with Density and
Perelman's Proof of the Poincaré Conjecture."
* Contributions by undergraduates. "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. "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."--Jacket