Geometric Measure Theory

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.

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