Morgan Geometric Measure Theory

A Beginner's Guide
2. Auflage 2014
ISBN: 978-1-4832-9664-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 175 Seiten, Web PDF

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.

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.

Autoren/Hrsg.

Morgan, Frank

Inhaltsverzeichnis

1;Front Cover;1
2;Geometric Measure Theory: A Beginner's Guide;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;8
6;CHAPTER 1. Geometric Measure Theory;12
6.1;1.1. Archetypical Problem.;12
6.2;1.2. Surfaces as Mappings.;12
6.3;1.3. The Direct Method.;14
6.4;1.4. Rectifiable Currents.;15
6.5;1.5. The Compactness Theorem.;17
6.6;1.6. Advantages of Rectifiable Currents.;17
6.7;1.7. The Regularity of Area-Minimizing Rectifiable Currents.;18
7;CHAPTER 2. Measures;20
7.1;2.1. Definitions.;20
7.2;2.2. Lebesgue Measure.;21
7.3;2.3. Hausdorff Measure [Federer, 2.10].;21
7.4;2.4. Integralgeometric Measure.;24
7.5;2.5. Densities [Federer, 2.9.12, 2.10.19].;25
7.6;2.6. Approximate Limits [Federer, 2.9.12].;26
7.7;2.7. Besicovitch Covering Theorem [Fédérer, 2.8.15; Besicovitch].;27
7.8;2.8. Corollary.;29
7.9;2.9. Corollary.;31
7.10;2.10. Corollary.;31
7.11;EXERCISES;31
8;CHAPTER 3. Lipschitz Functions and Rectifiable Sets;34
8.1;3.1. Lipschitz Functions.;34
8.2;3.2. Rademacher's Theorem [Federer, 3.1.6].;34
8.3;3.3. Approximation of a Lipschitz Function by a C1 Function [Federer, 3.1.15].;36
8.4;3.4. Lemma (Whitney's Extension Theorem) [Federer, 3.1.14].;36
8.5;3.5. Proposition [Federer, 2.10.11].;37
8.6;3.6. Jacobians.;38
8.7;3.7. The Area Formula [Federer, 3.2.3].;38
8.8;3.8. The Coarea Formula [Federer, 3.2.11].;40
8.9;3.9. Tangent Cones.;40
8.10;3.10. Rectifiable Sets [Federer, 3.2.14].;41
8.11;3.11. Proposition [cf. Federer (3.2.18, 3.2.19)].;42
8.12;3.12. Proposition [Federer, 3.2.19].;43
8.13;3.13. General Area-Coarea Formula [Federer, 3.2.22].;44
8.14;3.14. Product of measures [Federer, 3.2.23].;44
8.15;3.15. Orientation.;45
8.16;3.16. Crofton's Formula [Fédérer, 3.2.26].;45
8.17;3.17. Structure Theorem [Fédérer, 3.3.13].;46
8.18;EXERCISES;47
9;CHAPTER 4. Normal and Rectifiable Currents;50
9.1;4.1. Vectors and Differential Forms [Federer, Chapter 1 and 4.1].;50
9.2;4.2. Currents [Federer, 4.1.1, 4.1.7].;53
9.3;4.3. Important Spaces of Currents [Federer, 4.1.24, 4.1.22, 4.1.7, 4.1.5].;54
9.4;4.4. Theorem [Federer, 4.1.28].;59
9.5;4.5. Normal Currents [Federer, 4.1.7, 4.1.12].;60
9.6;4.6. Proposition [Fedérer, 4.1.17].;62
9.7;4.7. Theorem [Federer, 4.1.20].;62
9.8;4.8. Theorem [Federer, 4.1.23] .;64
9.9;4.9. Constancy Theorem [Federer, 4.1.31].;64
9.10;4.10. Cartesian Products.;65
9.11;4.11. Slicing [Federer, 4.2.1].;65
9.12;4.12. Lemma [Federer, 4.2.15].;69
9.13;EXERCISES;69
10;CHAPTER 5. The Compactness Theorem and the Existence of Area-Minimizing Surfaces;72
10.1;5.1. The Deformation Theorem [Federer, 4.2.9].;72
10.2;5.2. Corollary.;74
10.3;5.3. The Isoperimetric Inequality [Federer, 4.2.10].;75
10.4;5.4. The Closure Theorem [Federer, 4.2.16],;76
10.5;5.5. The Compactness Theorem [Federer, 4.2.17].;77
10.6;5.6. The Existence of Area-Minimizing Surfaces.;78
10.7;5.7. The Existence of Absolutely and Homologically Minimizing Surfaces in Manifolds [Federer, 5.1.6].;79
10.8;EXERCISES;79
11;CHAPTER 6. Examples of Area-Minimizing Surfaces;80
11.1;6.1. The Minimal Surface Equation [Federer, 5.4.18].;80
11.2;6.2. Remarks on Higher Dimensions.;87
11.3;6.3. Complex Analytic Varieties [Federer, 5.4.19].;87
11.4;6.4. Fundamental Theorem of Calibrations.;88
11.5;6.5. History of Calibrations (cf. Morgan [1, 2]).;88
11.6;EXERCISES;89
12;CHAPTER 7. The Approximation TheoremThe;90
12.1;7.1. The Approximation Theorem [Federer, 4.2.20].;90
13;CHAPTER 8. Survey of Regularity Results;94
13.1;8.1. Theorem [Fleming].;94
13.2;8.2. Theorem [Federer 2].;96
13.3;8.3. Theorem [Almgren, 3, 1983].;96
13.4;8.4. Boundary Regularity.;96
13.5;8.5. General Ambients and Other Integrands.;97
13.6;8.6. Theorem [Gonzalez, Massari, and Tamanini, Theorem 2].;98
13.7;EXERCISES;98
14;CHAPTER 9. Monotonicity and Oriented Tangent Cones;100
14.1;9.1. Locally Integral Flat Chains [Federer, 4.1.24, 4.3.16].;100
14.2;9.2. Monotonicity of the Mass Ratio.;101
14.3;9.3. Theorem [Federer, 5.4.3].;101
14.4;9.4. Corollary.;102
14.5;9.5. Corollary.;102
14.6;9.6. Corollary.;103
14.7;9.7. Oriented Tangent Cones [Federer, 4.3.16].;104
14.8;9.8. Theorem [Federer, 5.4.3(6)].;105
14.9;9.9. Theorem.;106
14.10;EXERCISES;107
15;CHAPTER 10. The Regularity of Area-Minimizing Hypersurfaces;108
15.1;10.1. Theorem.;108
15.2;10.2. Regularity for Area-Minimizing Hypersurfaces Theorem (Simons; see Federer [1, 5.4.15]).;112
15.3;10.3. Lemma [Federer, 5.4.6].;112
15.4;10.4. Maximum Principle.;112
15.5;10.5. Simons's Lemma [Federer, 5.4.14].;113
15.6;10.6. Lemma [Federer, 5.4.8, 5.4.9].;113
15.7;10.7. Remarks.;114
15.8;EXERCISES;116
16;CHAPTER 11. Flat Chains Modulo v, Varifolds, and (M,e,D)-Minimal Sets;118
16.1;11.1. Flat Chains Modulo v [Federer, 4.2.26].;118
16.2;11.2. Varifolds [Allard].;121
16.3;11.3. (M, e,ô)-Minimal Sets [Almgren 1].;122
16.4;EXERCISES;123
17;CHAPTER 12. Miscellaneous Useful Results;124
17.1;12.1. Morse-Sard-Federer Theorem.;124
17.2;12.2. Gauss-Green-De Giorgi-Federer Theorem.;124
17.3;12.3. Relative Homology [Federer, 4 . 4 ] .;126
17.4;12.4. Functions of Bounded Variation [Fédérer, 4.5.9; Giusti; Simon 2, §6].;128
17.5;12.5. General Parametric Integrands [Fédérer 1, 5.1].;129
18;CHAPTER 13. Soap Bubble Clusters;132
18.1;13.1. Planar Bubble Clusters.;134
18.2;13.2. Theory of Single Bubbles.;136
18.3;13.3. Cluster Theory.;139
18.4;13.4. Existence of Soap Bubble Clusters.;140
18.5;13.5. Lemma.;140
18.6;13.6. Lemma.;141
18.7;13.7. Sketch of Proof of Theorem 13.4.;141
18.8;13.8. Proposition.;142
18.9;13.9. Regularity of Soap Bubble Clusters in R3 [Taylor, 4].;144
18.10;13.10. Cluster Regularity in Higher Dimensions.;147
18.11;13.11. Minimizing Surface and Curve Energies.;147
18.12;13.12. Closing Remarks.;147
18.13;13.13. Kelvin disproved by Weaire and Phelan.;147
19;Solutions to Exercises;150
20;Bibliography;168
21;Index of Symbols;176
22;Name Index;180
23;Subject Index;182