Geometric Measure Theory

Geometric Measure Theory

A Beginner's Guide

5th Edition - April 7, 2016

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  • Author: Frank Morgan
  • Hardcover ISBN: 9780128044896
  • eBook ISBN: 9780128045275

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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.

Key Features

  • Focuses on core geometry rather than proofs, paving the way to fast and efficient insight into an extremely complex topic in geometric structures
  • Enables further study of more advanced topics and texts
  • Demonstrates in the simplest possible way how to relate concepts of geometric analysis by way of algebraic or topological techniques
  • Contains full topical coverage of The Log-Convex Density Conjecture
  • Comprehensively updated throughout


Graduate students and above with analysis backgrounds, researchers in differential geometry and geometric analysis

Table of Contents

  • 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

Product details

  • No. of pages: 272
  • Language: English
  • Copyright: © Academic Press 2016
  • Published: April 7, 2016
  • Imprint: Academic Press
  • Hardcover ISBN: 9780128044896
  • eBook ISBN: 9780128045275

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

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