Front Cover ; Dedication ; Geometric Measure Theory: A Beginner's Guide ; Copyright ; Contents; Preface; Part I: Basic Theory; Chapter 1: Geometric Measure Theory ; 1.1 Archetypical Problem; 1.2 Surfaces as Mappings; 1.3 The Direct Method; 1.4 Rectifiable Currents; 1.5 The Compactness Theorem; 1.6 Advantages of Rectifiable Currents; 1.7 The Regularity of Area-Minimizing Rectifiable Currents ; 1.8 More General Ambient Spaces; Chapter 2: Measures ; 2.1 Definitions; 2.2 Lebesgue Measure; 2.3 Hausdorff Measure ; 2.4 Integral-Geometric Measure; 2.5 Densities ; 2.6 Approximate Limits.

2.7 Besicovitch Covering Theorem 2.8 Corollary; 2.9 Corollary; 2.10 Corollary; Exercises; Chapter 3: Lipschitz Functions and Rectifiable Sets ; 3.1 Lipschitz Functions; 3.2 Rademacher's Theorem ; 3.3 Approximation of a Lipschitz Function by a C1 Funcation ; 3.4 Lemma (Whitney's Extension Theorem) ; 3.5 Proposition ; 3.6 Jacobians; 3.7 The Area Formula ; 3.8 The Coarea Formula ; 3.9 Tangent Cones; 3.10 Rectifiable Sets ; 3.11 Proposition ; 3.12 Proposition ; 3.13 General Area-Coarea Formula ; 3.14 Product of Measures ; 3.15 Orientation; 3.16 Crofton's Formula ; 3.17 Structure Theorem.

ExercisesChapter 4: Normal and Rectifiable Currents ; 4.1 Vectors and Differential Forms ; 4.2 Currents ; 4.3 Important Spaces of Currents ; 4.3A Mapping Currents; 4.3B Currents Representable by Integration; 4.4 Theorem ; 4.5 Normal Currents ; 4.6 Proposition ; 4.7 Theorem ; 4.8 Theorem ; 4.9 Constancy Theorem ; 4.10 Cartesian Products; 4.11 Slicing ; 4.12 Lemma ; 4.13 Proposition ; Exercises; Chapter 5: The Compactness Theorem and the Existence of Area-Minimizing Surfaces ; 5.1 The Deformation Theorem ; 5.2 Corollary; 5.3 The Isoperimetric Inequality ; 5.4 The Closure Theorem.

5.5 The Compactness Theorem 5.6 The Existence of Area-Minimizing Surfaces; 5.7 The Existence of Absolutely and Homologically Minimizing Surfaces in Manifolds ; Exercises; Chapter 6: Examples of Area-Minimizing Surfaces ; 6.1 The Minimal Surface Equation ; 6.2 Remarks on Higher Dimensions; 6.3 Complex Analytic Varieties ; 6.4 Fundamental Theorem of Calibrations; 6.5 History of Calibrations ; Exercises; Chapter 7: The Approximation Theorem ; 7.1 The Approximation Theorem ; Chapter 8: Survey of Regularity Results ; 8.1 Theorem ; 8.2 Theorem ; 8.3 Theorem ; 8.4 Boundary Regularity.

8.5 General Ambients, Volume Constraints, and Other IntegrandsExercises; Chapter 9: Monotonicity and Oriented Tangent Cones ; 9.1 Locally Integral Flat Chains ; 9.2 Monotonicity of the Mass Ratio; 9.3 Theorem ; 9.4 Corollary; 9.5 Corollary; 9.6 Corollary; 9.7 Oriented Tangent Cones ; 9.8 Theorem ; 9.9 Theorem; Exercises; Chapter 10: The Regularity of Area-Minimizing Hypersurfaces ; 10.1 Theorem; 10.2 Regularity for Area-Minimizing Hypersurfaces Theorem ; 10.3 Lemma ; 10.4 Maximum Principle; 10.5 Simons's Lemma ; 10.6 Lemma ; 10.7 Remarks; Exercises.