Contents
Preface vii
Part I: Basic Theory 1
1Geometric Measure Theory 3
2Measures 11
3Lipschitz Functions and Recti?able Sets 25
4Normal and Recti?able Currents 39
5The Compactness Theorem and the Existence of Area-Minimizing Surfaces 61
6Examples of Area-Minimizing Surfaces 69
7The Approximation Theorem 79
8Survey of Regularity Results 83
9Monotonicity and Oriented Tangent Cones 89
10The Regularity of Area-Minimizing Hypersurfaces 97
11Flat Chains Moduloν, Varifolds, and -Minimal Sets 105
12Miscellaneous Useful Results 111
Part II: Applications 119
13Soap Bubble Clusters 121
14Proof of Double Bubble Conjecture 143
15The Hexagonal Honeycomb and Kelvin Conjectures 159
16Immiscible Fluids and Crystals 173
17Isoperimetric Theorems in General Codimension 179
18Manifolds with Density and Perelman’s Proof of the Poincaré Conjecture 183
19Double Bubbles in Spheres, Gauss Space, and Tori 197
20The Log-Convex Density Theorem 205
Solutions to Exercises 213
Bibliography 235
Index of Symbols 255
Name Index 257
Subject Index 259
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