Geometric Measure Theory

Geometric Measure Theory

A Beginner's Guide

4th Edition - September 9, 2008

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  • Author: Frank Morgan
  • eBook ISBN: 9780080922409

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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

Product details

  • No. of pages: 264
  • Language: English
  • Copyright: © Academic Press 2008
  • Published: September 9, 2008
  • Imprint: Academic Press
  • eBook ISBN: 9780080922409

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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