Morgan Geometric Measure Theory

A Beginner's Guide
1. Auflage 2014
ISBN: 978-1-4832-7780-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 154 Seiten, Web PDF

Geometric Measure Theory: A Beginner's Guide provides information pertinent to the development of geometric measure theory. This book presents a few fundamental arguments and a superficial discussion of the regularity theory. Organized into 12 chapters, this book begins with an overview of the purpose and fundamental concepts of geometric measure theory. This text then provides the measure-theoretic foundation, including the definition of Hausdorff measure and covering theory. Other chapters consider the m-dimensional surfaces of geometric measure theory called rectifiable sets and introduce the two basic tools of the regularity theory of area-minimizing surfaces. This book discusses as well the fundamental theorem of geometric measure theory, which guarantees solutions to a wide class of variational problems in general dimensions. The final chapter deals with the basic methods of geometry and analysis in a generality that embraces manifold applications. This book is a valuable resource for graduate students, mathematicians, and research workers.

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;Geometrie 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;10
6.1;1.1. Archetypical Problem;10
6.2;1.2. Surfaces as a Mappings;10
6.3;1.3. The Direct Method;11
6.4;1.4. Rectifiable Currents;14
6.5;1.5. The Compactness Theorem;15
6.6;1.6. Advantages of Rectifiable Currents;15
6.7;1.7. The Regularity of Area-minimizing Rectifiable Currents;15
7;CHAPTER 2. Measures;16
7.1;2.1. Definitions;16
7.2;2.2. Lebesgue Measure;17
7.3;2.3. Hausdorff Measure [GMT 2.10];17
7.4;2.4. Integralgeometric Measure;19
7.5;2.5. Densities [GMT 2.9.12, 2.10.19];21
7.6;2.6. Approximate Limits [GMT 2.9.12];22
7.7;2.7. Besicovitch Covering Theorem [GMT 2.8.15];23
7.8;2.8.
Corollary. Hn = Ln on Rn;26
7.9;2.9. Corollary;27
7.10;2.10. Corollary;28
7.11;EXERCISES;28
8;CHAPTER 3. Lipschitz Functions and Rectifiable Sets;30
8.1;3.1. Lipschitz Functions;30
8.2;3.2. Rademacher's Theorem [GMT 3.1.6];30
8.3;3.3. Approximation of a Lipschitz Function by a
C1 Function [GMT 3.1.15].;32
8.4;3.4.
Lemma (Whitney's Extention Theorem) [GMT 3.1.14];32
8.5;3.5. Proposition [GMT 2.10.11];33
8.6;3.6. Jacobians;34
8.7;3.7. The Area Formula [GMT 3.2, 3];34
8.8;3.8. The Coarea Formula [GMT 3.2.11];36
8.9;3.9. Tangent Cones;37
8.10;3.10. Rectifiable Sets [GMT 3.2.14];37
8.11;3.11. Proposition [cf. GMT 3.2.18, 3.2.29];39
8.12;3.12. Proposition [GMT 3.2.19];39
8.13;3.13. General Area-coarea Formula [GMT 3.2.22];40
8.14;3.14. Product of measures [GMT 3.2.23];41
8.15;3.16. Crofton's Formula [GMT 3.2.26];41
8.16;3.17. Structure Theorem [GMT 3.3.13];42
8.17;EXERCISES;44
9;CHAPTER 4. Normal and Rectifiable Currents;46
9.1;4.1. Vectors and Differential Forms [GMT, Chapter 1 and 4.1];46
9.2;4.2. Currents [GMT 4.1.1, 4.1.7];49
9.3;4.3. Important Spaces of Currents [GMT 4.1.24, 4.1.22, 4.1.7, 4.1.5];50
9.4;4.4. Theorem [GMT 4.1.28];55
9.5;4.5. Normal Currents [GMT 4.1.7, 4.1.12];56
9.6;4.6. Proposition [GMT 4.1.17];58
9.7;4.7. Theorem [GMT 4.1.20];58
9.8;4.8. Theorem [GMT 4.1.23];60
9.9;4.9. Constancy Theorem [GMT 4.1.31];61
9.10;4.10. Cartesian Products;62
9.11;4.11. Slicing [GMT 4.2.1];62
9.12;4.12. Lemma [GMT 4.2.15];65
9.13;EXERCISES;65
10;CHAPTER 5. The Compactness Theorem and the Existence of Area-Minimizing Surfaces;68
10.1;5.1. The Deformation Theorem [GMT 4.2.9];68
10.2;5.2. Corollary;71
10.3;5.3. The Isoperimetric Inequality [GMT 4.2.10];71
10.4;5.4. The Closure Theorem [GMT 4.2.16];71
10.5;5.5. The Compactness Theorem [GMT 4.2.17];73
10.6;5.6. The Existence of Area-minimizing Surfaces;74
10.7;5.7. The Existence of Absolutely and Homologically Minimizing Surfaces in Manifolds [GMT 5.1.6];75
10.8;EXERCISES;75
11;CHAPTER 6. Examples of Area-Minimizing Surfaces;76
11.1;6.1. The Minimal Surface Equation [GMT 5.4.18];76
11.2;6.2. Remarks on Higher Dimensions;82
11.3;6.3. Complex Analytic Varieties [GMT 5.4.19];83
11.4;EXERCISES;84
12;CHAPTER 7. The Approximation Theorem;86
12.1;7.1. The Approximation Theorem [GMT 4.1.24];86
13;CHAPTER 8. Survey of Regularity Results;90
13.1;8.1. Theorem;90
13.2;8.2. Theorem [Fe];91
13.3;8.3. Theorem;92
13.4;8.4. Boundary Regularity;93
13.5;8.5. General Ambients and Other Integrands;93
13.6;EXERCISES;94
14;CHAPTER 9. Monotonicity and Oriented Tangent Cones;96
14.1;9.1. Locally Integral Flat Chains [GMT 4.1.24, 4.3.16];96
14.2;9.2. Monotonicity of the Mass Ratio;97
14.3;9.3. Theorem [GMT 5.4.3];98
14.4;9.4. Corollary;98
14.5;9.5. Corollary;98
14.6;9.7. Oriented Tangent Cones [GMT 4.3.16];100
14.7;9.8. Theorem [GMT 5.4.36];101
14.8;EXERCISES;103
15;CHAPTER 10. The Regularity of Area-Minimizing Hypersurfaces;106
15.1;10.1. Theorem;106
15.2;10.2. Regularity for Area-minimizing Hypersurfaces Theorem;110
15.3;10.3. Lemma [GMT 5.4.6];110
15.4;10.4. Maximum Principle;111
15.5;10.5. Simons's Lemma [GMT 5.4.14];111
15.6;10.6. Lemma [GMT 5.4.8, 5.4.9];111
15.7;10.7. Remarks;112
15.8;EXERCISES;114
16;CHAPTER 11. Flat Chains Modulo v, Varifolds, and (M, e, d)-Minimal
Sets;116
16.1;11.1. Flat Chains Modulo
v [GMT 4.2.26];116
16.2;11.2. Varifolds [All];118
16.3;11.3. (M, e,
d)-minimal Sets [Alm 1];119
16.4;EXERCISES;120
17;CHAPTER 12. Miscellaneous Useful Results;122
17.1;12.1. Morse-Sard-Federer Theorem;122
17.2;12.2. Gauss-Green-Federer Theorem;123
17.3;12.3. Relative homology [GMT 4.4];124
17.4;12.4. Functions of Bounded Variation [GMT 4.5.9], [G], [SL1 §6];125
18;Solutions to Exercises;128
18.1;Chapter 2;128
18.2;Chapter 3;130
18.3;Chapter 4;131
18.4;Chapter 5;136
18.5;Chapter 6;137
18.6;Chapter 8;138
18.7;Chapter 9;140
18.8;Chapter 10;142
18.9;Chapter 11;143
19;Bibliography;144
20;Index of Symbols;148
21;Name Index;150
22;Subject Index;152