Geometric Measure Theory: A Beginners Guide

Geometric Measure Theory: A Beginner's Guide, Fifth Edition provides the framework readers need to understand the structure of a crystal, a soap bubble cluster, or a universe.

The book is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Brevity, clarity, and scope make this classic book an excellent introduction to more complex ideas from geometric measure theory and the calculus of variations for beginning graduate students and researchers.

Morgan emphasizes geometry over proofs and technicalities, providing a fast and efficient insight into many aspects of the subject, with new coverage to this edition including topical coverage of the Log Convex Density Conjecture, a major new theorem at the center of an area of mathematics that has exploded since its appearance in Perelman's proof of the Poincaré conjecture, and new topical coverage of manifolds taking into account all recent research advances in theory and applications.

Focuses on core geometry rather than proofs, paving the way to fast and efficient insight into an extremely complex topic in geometric structuresEnables further study of more advanced topics and textsDemonstrates in the simplest possible way how to relate concepts of geometric analysis by way of algebraic or topological techniquesContains full topical coverage of The Log-Convex Density ConjectureComprehensively updated throughout

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,E,)-Minimal Sets 12. Miscellaneous Useful Results 13. Soap Bubble Clusters 14. Proof of Double Bubble Conjecture 15. The Hexagonal Honeycomb and Kelvin Conjectures 16. Immiscible Fluids and Crystals 17. Isoperimetric Theorems in General Codimension 18. Manifolds with Density and Perelman's Proof of the Poincaré Conjecture 19. Double Bubbles in Spheres, Gauss Space, and Tori 20. The Log-Convex Density Conjecture 21. Solutions to Exercises