Geometric Integration Theory

By: Steven G. Krantz, Harold R. Parks

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Hardcover | 12 August 2008

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356 Pages

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Geometric measure theory has roots going back to ancient Greek mathematics, for considerations of the isoperimetric problem (to ?nd the planar domain of given perimeter having greatest area) led naturally to questions about spatial regions and boundaries. In more modern times, the Plateau problem is considered to be the wellspring of questions in geometric measure theory. Named in honor of the nineteenth century Belgian physicist Joseph Plateau, who studied surface tension phenomena in general, andsoap?lmsandsoapbubblesinparticular,thequestion(initsoriginalformulation) was to show that a ?xed, simple, closed curve in three-space will bound a surface of the type of a disk and having minimal area. Further, one wishes to study uniqueness for this minimal surface, and also to determine its other properties. Jesse Douglas solved the original Plateau problem by considering the minimal surfacetobeaharmonicmapping(whichoneseesbystudyingtheDirichletintegral). For this work he was awarded the Fields Medal in 1936. Unfortunately, Douglas's methods do not adapt well to higher dimensions, so it is desirable to ?nd other techniques with broader applicability. Enter the theory of currents. Currents are continuous linear functionals on spaces of differential forms.

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From the reviews:

"This is a graduate textbook with the main purpose of introducing geometric measure theory through the notion of currents. ... One of the most important features of this text is that it is self-contained ... . The book also contains an Appendix ... as well as extended list of references, making it a good text for a graduate course, as well as for an independent or self study." (Mihaela Poplicher, The Mathematical Association of America, March, 2009)

"The book under review succeeds in giving a complete and readable introduction to geometric measure theory. It can be used by students willing to learn this beautiful theory or by teachers as a basis for a one- or two-semester course." (Andreas Bernig, Mathematical Reviews, Issue 2009 m)

"The authors present main fields of applications, namely the isoperimetric problem and the regularity of minimal currents. The exposition is detailed and very well organized and therefore the book should be quite accessible for graduate students." (R. Steinbauer, Monatshefte fuer Mathematik, Vol. 162 (3), March, 2011)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Calculus Algebra Geometry Differential & Riemannian Geometry

Preface	p. xi
Basics	p. 1
Smooth Functions	p. 1
Measures	p. 6
Lebesgue Measure	p. 10
Integration	p. 12
Measurable Functions	p. 12
The Integral	p. 14
Lebesgue Spaces	p. 20
Product Measures and the Fubini-Tonelli Theorem	p. 21
The Exterior Algebra	p. 22
The Generalized Pythagorean Theorem	p. 25
The Hausdorff Distance and Steiner Symmetrization	p. 33
Borel and Suslin Sets	p. 42
Caratheodory's Construction and Lower-Dimensional Measures	p. 53
The Basic Definition	p. 53
Hausdorff Measure and Spherical Measure	p. 55
A Measure Based on Parallelepipeds	p. 56
Projections and Convexity	p. 57
Other Geometric Measures	p. 58
Summary	p. 59
The Densities of a Measure	p. 61
A One-Dimensional Example	p. 63
Caratheodory's Construction and Mappings	p. 64
The Concept of Hausdorff Dimension	p. 67
Some Cantor Set Examples	p. 69
Basic Examples	p. 70
Some Generalized Cantor Sets	p. 72
Cantor Sets in Higher Dimensions	p. 73
Invariant Measures and the Construction of Haar Measure	p. 77
The Fundamental Theorem	p. 78
Haar Measure for the Orthogonal Group and the Grassmannian	p. 84
Remarks on the Manifold Structure of G(N, M)	p. 88
Covering Theorems and the Differentiation of Integrals	p. 91
Wiener's Covering Lemma and Its Variants	p. 91
The Besicovitch Covering Theorem	p. 99
Decomposition and Differentiation of Measures	p. 109
The Riesz Representation Theorem	p. 115
Maximal Functions Redux	p. 122
Analytical Tools: The Area Formula, the Coarea Formula, and Poincare Inequalities	p. 125
The Area Formula	p. 125
Linear Maps	p. 126
C[superscript 1] Functions	p. 132
Rademacher's Theorem	p. 134
The Coarea Formula	p. 137
Measure Theory of Lipschitz Maps	p. 140
Proof of the Coarea Formula	p. 142
The Area and Coarea Formulas for C[superscript 1] Submanifolds	p. 143
Rectifiable Sets	p. 148
Poincare Inequalities	p. 151
The Calculus of Differential Forms and Stokes's Theorem	p. 159
Differential Forms and Exterior Differentiation	p. 159
Stokes's Theorem	p. 164
Introduction to Currents	p. 173
A Few Words about Distributions	p. 174
The Definition of a Current	p. 177
Constructions Using Currents and the Constancy Theorem	p. 183
Further Constructions with Currents	p. 189
Products of Currents	p. 189
The Pushforward	p. 190
The Homotopy Formula	p. 193
Applications of the Homotopy Formula	p. 193
Rectifiable Currents with Integer Multiplicity	p. 195
Slicing	p. 204
The Deformation Theorem	p. 211
Proof of the Unscaled Deformation Theorem	p. 217
Applications of the Deformation Theorem	p. 222
Currents and Calculus of Variations	p. 225
Proof of the Compactness Theorem	p. 225
Integer-Multiplicity 0-Currents	p. 226
A Rectifiability Criterion for Currents	p. 231
MBV Functions	p. 232
The Slicing Lemma	p. 237
The Density Lemma	p. 238
Completion of the Proof of the Compactness Theorem	p. 240
The Flat Metric	p. 241
Existence of Currents Minimizing Variational Integrals	p. 244
Minimizing Mass	p. 244
Other Integrands and Integrals	p. 245
Density Estimates for Minimizing Currents	p. 250
Regularity of Mass-Minimizing Currents	p. 255
Preliminaries	p. 256
The Height Bound and Lipschitz Approximation	p. 262
Currents Defined by Integrating over Graphs	p. 269
Estimates for Harmonic Functions	p. 272
The Main Estimate	p. 286
The Regularity Theorem	p. 303
Epilogue	p. 308
Appendix	p. 311
Transfinite Induction	p. 311
Dual Spaces	p. 313
Line Integrals	p. 316
Pullbacks and Exterior Derivatives	p. 319
References	p. 323
Index of Notation	p. 329
Index	p. 335
Table of Contents provided by Ingram. All Rights Reserved.

Geometric Integration Theory

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