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Geometric Measure Theory - 4th Edition - ISBN: 9780123744449, 9780080922409

Geometric Measure Theory

4th Edition

A Beginner's Guide

Author: Frank Morgan
eBook ISBN: 9780080922409
Imprint: Academic Press
Published Date: 9th September 2008
Page Count: 264
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Geometric Measure Theory, Fourth Edition, is an excellent text for introducing ideas from geometric measure theory and the calculus of variations to beginning graduate students and researchers.

This updated edition contains abundant illustrations, examples, exercises, and solutions; and the latest results on soap bubble clusters, including a new chapter on Double Bubbles in Spheres, Gauss Space, and Tori. It also includes a new chapter on Manifolds with Density and Perelman's Proof of the Poincaré Conjecture.

This text is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Morgan emphasizes geometry over proofs and technicalities providing a fast and efficient insight into many aspects of the subject.

Key Features

New to the 4th edition:
* Abundant illustrations, examples, exercises, and solutions.
* The latest results on soap bubble clusters,
including a new chapter on "Double Bubbles in
Spheres, Gauss Space, and Tori."
* A new chapter on "Manifolds with Density and
Perelman's Proof of the Poincaré Conjecture."
* Contributions by undergraduates.


Advanced graduate students and researchers in mathematics.

Table of Contents

Geometric Measure Theory
Lipschitz Functions and Rectifiable Sets
Normal and Rectifiable Currents
The Compactness Theorem and the Existence of Area-Minimizing Surfaces
Examples of Area-Minimizing Surfaces
The Approximation Theorem
Survey of Regularity Results
Monotonicity and Oriented Tangent Cones
The Regularity of Area-Minimizing Hypersurfaces
Flat Chains Modulo v, Varifolds, and (M,E,)-Minimal Sets
Miscellaneous Useful Results
Soap Bubble Clusters
Proof of Double Bubble Conjecture
The Hexagonal Honeycomb and Kelvin Conjectures
Immiscible Fluids and Crystals
Isoperimetric Theorems in General Codimension
Manifolds with Density and Perelman's Proof of the Poincaré Conjecture
Double Bubbles in Spheres, Gauss Space, and Tori
Solutions to Exercises


No. of pages:
© Academic Press 2008
9th September 2008
Academic Press
eBook ISBN:

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


“The text is simply unique. It doesn't compare to any other because its goals are different. It cannot be used as the only source of information for learning GMT, yet learning this subject without owning a copy of this book would be ridiculous since it gives a fast and efficient insight in many aspects of the theory.”
-Thierry De Pauw, niversite catholique de Louvain, Belgium

“The book is unique in its format and exposition. Without it, it would be difficult to get in touch with the subject. It paves the way to more advanced books. All other books on the market about this subject are rather technical and difficult to read for an inexperienced student.”
-Stefan Wenger, Courant Institute of Math, New York University

Ratings and Reviews