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

✍ Scribed by Frank Morgan

• Publisher: Academic Press
• Year: 2009
• Tongue: English
• Leaves: 259
• Edition: 4
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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