Author(s): Leon Simon
Year: 1983
Language: English
Commentary: complete
CONTENTS
INTRODUCTION
NOTATION
ERRATA
CHAPTER 1 : PRELIMINARY MEASURE THEORY
1. Basic Notions
2. Hausdorff Measure
3. Densities
4. Radon Measures
CHAPTER 2 : SOME FURTHER PRELIMINARIES FROM ANALYSIS
5. Lipschitz Functions
6. BV Functions
7. Submanifolds of R^{n+k}
8. The Area Formula
9. First and Second Variation Formulae
10. The Co-area Formula
CHAPTER 3 : COUNTABLY n-RECTIFIABLE SETS
11. Basic Notions, Tangent Properties
12. Gradients, Jacobians, Area, Co-area
13. The Structure Theorem
14. Sets of Locally Finite Perimeter
CHAPTER 4 : THEORY OF RECTIFIABLE n-VARIFOLDS
15. Basic Definitions and Properties
16. First Variation
17. Monotonicity Formulae and Basic Consequences
18. Poincare and Sobolev Inequalities
19. Miscellaneous Additional Consequences of the Monotonicity Formulae
CHAPTER 5 : THE ALLARD REGULARITY THEOREM
20. Lipschitz Approximation
21. Approximation by Harmonic Functions
22. The Tilt-Excess Decay Lemma
23. Main Regularity Theorem: First Version
24. Main Regularity Theorem: Second Version
CHAPTER 6 : CURRENTS
25. Preliminaries: Vectors, Co-vectors, and Forms
26. General Currents
27. Integer Multiplicity Rectifiable Currents
28. Slicing
29. The Deformation Theorem
30. Applications of the Deformation Theorem
31. The Flat Metric Topology
32. Rectifiability Theorem, and proof of the compactness theorem
CHAPTER 7 : AREA MINIMIZING CURRENTS
33. Basic Concepts
34. Existence and Compactness Results.
35. Tangent Cones and Densities
36. Some Regularity Results (Arbitrary Codimension)
37. Codimension 1 Theory
CHAPTER 8 : THEORY OF GENERAL VARIFOLDS
38. Basics, First Rectifiability Theorem
39. First Variation
40. Monotonicity and Consequences
41. Constancy Theorem
42. Varifold Tangents and Rectifiability Theorem
APPENDIX A : A GENERAL REGULARITY THEOREM
APPENDIX B NON-EXISTENCE OF STABLE MINIMAL CONES, n <_ 6.
REFERENCES
