Geometric Measure Theory

Geometric measure theory is the mathematical framework for the study of crystal growth, clusters of soap bubbles, and similar structures involving minimization of energy. Morgan emphasizes geometry over proofs and technicalities, and includes a bibliography and abundant illustrations and examples. This Second Edition features a new chapter on soap bubbles as well as updated sections addressing volume constraints, surfaces in manifolds, free boundaries, and Besicovitch constant results. The text will introduce newcomers to the field and appeal to mathematicians working in the field.

Preface	p. vii
Geometric Measure Theory	p. 1
Measures	p. 7
Lipschitz Functions and Rectifiable Sets	p. 21
Normal and Rectifiable Currents	p. 35
The Compactness Theorem and the Existence of Area-Minimizing Surfaces	p. 59
Examples of Area-Minimizing Surfaces	p. 67
The Approximation Theorem	p. 77
Survey of Regularity Results	p. 81
Monotonicity and Oriented Tangent Cones	p. 87
The Regularity of Area-Minimizing Hypersurfaces	p. 97
Flat Chains Modulo [upsilon], Varifolds, and (M, [varepsilon], [delta])-Minimal Sets	p. 105
Miscellaneous Useful Results	p. 113
Soap Bubble Clusters	p. 121
Proof of Double Bubble Conjecture	p. 141
The Hexagonal Honeycomb and Kelvin Conjectures	p. 157
Immiscible Fluids and Crystals	p. 173
Isoperimetric Theorems in General Codimension	p. 181
Solutions to Exercises	p. 185
Bibliography	p. 203
Index of Symbols	p. 217
Name Index	p. 221
Subject Index	p. 223
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Geometric Measure Theory

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