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 Modulo, 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 Perelmans 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