Geometric Measure Theory - A Beginner's Guide

Preface......Page 1
1 Geometric Measure Theory......Page 4
2 Measures......Page 10
3 Lipschitz Functions and Rectifiable Sets......Page 23
4 Normal and Rectifiable Currents......Page 37
5 The Compactness Theorem and the Existence of Area-Minimizing Surfaces......Page 60
6 Examples of Area-Minimizing Surfaces......Page 68
7 The Approximation Theorem......Page 78
8 Survey of Regularity Results......Page 81
9 Monotonicity and Oriented Tangent Cones......Page 87
10 The Regularity of Area-Minimizing Hypersurfaces......Page 96
11 Flat Chains Modulo
, Varifolds, and (M, ε, υ)-Minimal Sets......Page 104
12 Miscellaneous Useful Results......Page 111
13 Soap Bubble Clusters......Page 119
14 Proof of Double Bubble Conjecture......Page 138
15 The Hexagonal Honeycomb and Kelvin Conjectures......Page 154
16 Immiscible Fluids and Crystals......Page 169
17 Isoperimetric Theorems in General Codimension......Page 176
Solutions to Exercises......Page 180
Bibliography......Page 197
Index of Symbols......Page 210
Name Index......Page 213
Subject Index......Page 215