Geometric Measure Theory
1st Edition
A Beginner's Guide
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Description
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.
Table of Contents
Preface
1. Geometric Measure Theory
2. Measures
3. Lipschitz Functions and Rectifiable Sets
4. Normal and Rectifiable Currents
5. The Compactness Theorem and the Existence of Area- Minimizing Surfaces
6. Examples of Area-Minimizing Surfaces
7. The Approximation Theorem
8. Survey of Regularity Results
9. Monotonicity and Oriented Tangent Cones
10. The Regularity of Area-Minimizing Hypersurfaces
11. Flat Chains Modulo v, Varifolds, and (M, ε, δ)-Minimal Sets
12. Miscellaneous Useful Results
Solutions to Exercises
Bibliography
Index of Symbols
Name Index
Subject Index
Details
- No. of pages:
- 154
- Language:
- English
- Copyright:
- © Academic Press 1988
- Published:
- 28th March 1988
- Imprint:
- Academic Press
- eBook ISBN:
- 9781483277806
About the Author
Frank Morgan
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.
Affiliations and Expertise
Williams College, Williamstown, MA, USA