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Geometric Integration Theory

Steven G. Krantz and Harold R. Parks
Publisher: 
Birkhäuser
Publication Date: 
2008
Number of Pages: 
339
Format: 
Hardcover
Series: 
Cornerstones
Price: 
69.95
ISBN: 
978-0-8176-4676-9
Category: 
Textbook
[Reviewed by
Mihaela Poplicher
, on
03/23/2009
]

This is a graduate textbook with the main purpose of introducing geometric measure theory through the notion of currents. A current is defined intuitively as a differential form with coefficients that are distributions or as a continuous linear functional on a space of differential forms.

Even from this very brief summary, it is obvious that there is considerable background required for a successful reading of this book. One of the most important features of this text is that it is self-contained, not an easy task for an area that applies techniques to complex geometry, partial differential equations, harmonic analysis, differential geometry.

The background material provided in the text in order to make it self-contained includes:

  • Measure theory;
  • Caratheodory’s Construction and Lower Dimensional Measures;
  • Invariant Measures and Haar Measure;
  • Covering Theorems and Differentiation of Integrals;
  • The Area Formula, the Coarea Formula, and Poincare Inequalities;
  • Calculus of Differential Forms and Stoke’s Theorem.

The remaining chapters of the book provide a detailed treatment of currents:

  • Introduction to Currents  (beginning with a look at distributions);
  • Currents and the Calculus of Variations;
  • Regularity of Mass-Minimizing Currents.

The treatment of geometric measure theory as outlined above (following the development by Federer and Fleming who first used currents in the 1950s and then used them in 1960 to solve the general Plateau problem, the wellspring of questions in geometric measure theory) constitutes a good introduction of the graduate student to key ideas in the subject.

The book also contains an Appendix (including Transfinite Induction, Line Integrals, Dual Spaces, and Pullbacks) as well as an extended list of references, making it a good text for a graduate course, as well as for an independent or self study.


Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.

 

Preface.- Basics.- Carathéodory’s Construction and Lower-Dimensional Measures.- Invariant Meaures and the Construction of Haar Meaure.- Covering Theorems and the Differentiation of Integrals.- Analytical Tools: the Area Formula, the Coarea Formula, and Poincaré Inequalities.- The Calculus of Differential Forms and Stokes’s Theorem.- Introduction to Currents.- Currents and the Calculus of Variations.- Regularity of Mass-Minimizing Currents.- Appendix.-References.- Index of Notation.- Index.

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