Geometric Measure Theory (eBook)
175 Seiten
Elsevier Science (Verlag)
978-1-4832-9664-7 (ISBN)
Frank Morgan is the Dennis Meenan '54 Third Century Professor of Mathematics at Williams College. He obtained his B.S. from MIT and his M.S. and Ph.D. from Princeton University. His research interest lies in minimal surfaces, studying the behavior and structure of minimizers in various settings. He has also written Riemannian Geometry: A Beginner's Guide, Calculus Lite, and most recently The Math Chat Book, based on his television program and column on the Mathematical Association of America Web site.
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.
Front Cover 1
Geometric Measure Theory: A Beginner's Guide 4
Copyright Page 5
Table of Contents 6
Preface 8
CHAPTER 1. Geometric Measure Theory 12
1.1. Archetypical Problem. 12
1.2. Surfaces as Mappings. 12
1.3. The Direct Method. 14
1.4. Rectifiable Currents. 15
1.5. The Compactness Theorem. 17
1.6. Advantages of Rectifiable Currents. 17
1.7. The Regularity of Area-Minimizing Rectifiable Currents. 18
CHAPTER 2. Measures 20
2.1. Definitions. 20
2.2. Lebesgue Measure. 21
2.3. Hausdorff Measure [Federer, 2.10]. 21
2.4. Integralgeometric Measure. 24
2.5. Densities [Federer, 2.9.12, 2.10.19]. 25
2.6. Approximate Limits [Federer, 2.9.12]. 26
2.7. Besicovitch Covering Theorem [Fédérer, 2.8.15 Besicovitch].
2.8. Corollary. 29
2.9. Corollary. 31
2.10. Corollary. 31
EXERCISES 31
CHAPTER 3. Lipschitz Functions and Rectifiable Sets 34
3.1. Lipschitz Functions. 34
3.2. Rademacher's Theorem [Federer, 3.1.6]. 34
3.3. Approximation of a Lipschitz Function by a C1 Function [Federer, 3.1.15]. 36
3.4. Lemma (Whitney's Extension Theorem) [Federer, 3.1.14]. 36
3.5. Proposition [Federer, 2.10.11]. 37
3.6. Jacobians. 38
3.7. The Area Formula [Federer, 3.2.3]. 38
3.8. The Coarea Formula [Federer, 3.2.11]. 40
3.9. Tangent Cones. 40
3.10. Rectifiable Sets [Federer, 3.2.14]. 41
3.11. Proposition [cf. Federer (3.2.18, 3.2.19)]. 42
3.12. Proposition [Federer, 3.2.19]. 43
3.13. General Area-Coarea Formula [Federer, 3.2.22]. 44
3.14. Product of measures [Federer, 3.2.23]. 44
3.15. Orientation. 45
3.16. Crofton's Formula [Fédérer, 3.2.26]. 45
3.17. Structure Theorem [Fédérer, 3.3.13]. 46
EXERCISES 47
CHAPTER 4. Normal and Rectifiable Currents 50
4.1. Vectors and Differential Forms [Federer, Chapter 1 and 4.1]. 50
4.2. Currents [Federer, 4.1.1, 4.1.7]. 53
4.3. Important Spaces of Currents [Federer, 4.1.24, 4.1.22, 4.1.7, 4.1.5]. 54
4.4. Theorem [Federer, 4.1.28]. 59
4.5. Normal Currents [Federer, 4.1.7, 4.1.12]. 60
4.6. Proposition [Fedérer, 4.1.17]. 62
4.7. Theorem [Federer, 4.1.20]. 62
4.8. Theorem [Federer, 4.1.23] . 64
4.9. Constancy Theorem [Federer, 4.1.31]. 64
4.10. Cartesian Products. 65
4.11. Slicing [Federer, 4.2.1]. 65
4.12. Lemma [Federer, 4.2.15]. 69
EXERCISES 69
CHAPTER 5. The Compactness Theorem and the Existence of Area-Minimizing Surfaces 72
5.1. The Deformation Theorem [Federer, 4.2.9]. 72
5.2. Corollary. 74
5.3. The Isoperimetric Inequality [Federer, 4.2.10]. 75
5.4. The Closure Theorem [Federer, 4.2.16], 76
5.5. The Compactness Theorem [Federer, 4.2.17]. 77
5.6. The Existence of Area-Minimizing Surfaces. 78
5.7. The Existence of Absolutely and Homologically Minimizing Surfaces in Manifolds [Federer, 5.1.6]. 79
EXERCISES 79
CHAPTER 6. Examples of Area-Minimizing Surfaces 80
6.1. The Minimal Surface Equation [Federer, 5.4.18]. 80
6.2. Remarks on Higher Dimensions. 87
6.3. Complex Analytic Varieties [Federer, 5.4.19]. 87
6.4. Fundamental Theorem of Calibrations. 88
6.5. History of Calibrations (cf. Morgan [1, 2]). 88
EXERCISES 89
CHAPTER 7. The Approximation TheoremThe 90
7.1. The Approximation Theorem [Federer, 4.2.20]. 90
CHAPTER 8. Survey of Regularity Results 94
8.1. Theorem [Fleming]. 94
8.2. Theorem [Federer 2]. 96
8.3. Theorem [Almgren, 3, 1983]. 96
8.4. Boundary Regularity. 96
8.5. General Ambients and Other Integrands. 97
8.6. Theorem [Gonzalez, Massari, and Tamanini, Theorem 2]. 98
EXERCISES 98
CHAPTER 9. Monotonicity and Oriented Tangent Cones 100
9.1. Locally Integral Flat Chains [Federer, 4.1.24, 4.3.16]. 100
9.2. Monotonicity of the Mass Ratio. 101
9.3. Theorem [Federer, 5.4.3]. 101
9.4. Corollary. 102
9.5. Corollary. 102
9.6. Corollary. 103
9.7. Oriented Tangent Cones [Federer, 4.3.16]. 104
9.8. Theorem [Federer, 5.4.3(6)]. 105
9.9. Theorem. 106
EXERCISES 107
CHAPTER 10. The Regularity of Area-Minimizing Hypersurfaces 108
10.1. Theorem. 108
10.2. Regularity for Area-Minimizing Hypersurfaces Theorem (Simons see Federer [1, 5.4.15]).
10.3. Lemma [Federer, 5.4.6]. 112
10.4. Maximum Principle. 112
10.5. Simons's Lemma [Federer, 5.4.14]. 113
10.6. Lemma [Federer, 5.4.8, 5.4.9]. 113
10.7. Remarks. 114
EXERCISES 116
CHAPTER 11. Flat Chains Modulo v, Varifolds, and (M,e,D)-Minimal Sets 118
11.1. Flat Chains Modulo v [Federer, 4.2.26]. 118
11.2. Varifolds [Allard]. 121
11.3. (M, e,ô)-Minimal Sets [Almgren 1]. 122
EXERCISES 123
CHAPTER 12. Miscellaneous Useful Results 124
12.1. Morse-Sard-Federer Theorem. 124
12.2. Gauss-Green-De Giorgi-Federer Theorem. 124
12.3. Relative Homology [Federer, 4 . 4 ] . 126
12.4. Functions of Bounded Variation [Fédérer, 4.5.9 Giusti
12.5. General Parametric Integrands [Fédérer 1, 5.1]. 129
CHAPTER 13. Soap Bubble Clusters 132
13.1. Planar Bubble Clusters. 134
13.2. Theory of Single Bubbles. 136
13.3. Cluster Theory. 139
13.4. Existence of Soap Bubble Clusters. 140
13.5. Lemma. 140
13.6. Lemma. 141
13.7. Sketch of Proof of Theorem 13.4. 141
13.8. Proposition. 142
13.9. Regularity of Soap Bubble Clusters in R3 [Taylor, 4]. 144
13.10. Cluster Regularity in Higher Dimensions. 147
13.11. Minimizing Surface and Curve Energies. 147
13.12. Closing Remarks. 147
13.13. Kelvin disproved by Weaire and Phelan. 147
Solutions to Exercises 150
Bibliography 168
Index of Symbols 176
Name Index 180
Subject Index 182
| Erscheint lt. Verlag | 19.5.2014 |
|---|---|
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| Technik | |
| ISBN-10 | 1-4832-9664-4 / 1483296644 |
| ISBN-13 | 978-1-4832-9664-7 / 9781483296647 |
| Haben Sie eine Frage zum Produkt? |
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