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Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants . 98 4.2 The Besicovitch Covering Theorem . 106 4.3 Decomposition and Differentiation of Measures . 111 4.4 Maximal Functions Redux . 119 5 Analytical Tools: the Area Formula, the Coarea Formula, and Poincar´eInequalities 121 5.1 The Area Formula . 121 5.1.1 Linear Maps . 122 5.1.2 C1 Functions.......................129 5.1.3 Rademacher’s Theorem . 131 5.2 The Coarea Formula . 135 5.2.1 Measure Theory of Lipschitz Maps . 139 5.2.2 Proof of the Coarea Formula . 141 5.3 The Area and Coarea Formulas for C1 Submanifolds . 143 5.4 Rectifiable Sets . 148 5.5 Poincar´e Inequalities . 152 6 The Calculus of Differential Forms and Stokes’s Theorem 161 6.1 Differential Forms and Exterior Differentiation . 161 6.2 Stokes’s Theorem . 167 7 Introduction to Currents 177 7.1 A Few Words about Distributions . 178 7.2 The Definition of a Current . 182 7.3 Constructions Using Currents and the Constancy Theorem . 189 7.4 Further Constructions with Currents . 196 7.4.1 Products of Currents . 196 CONTENTS iii 7.4.2 The Push-Forward . 197 7.4.3 The Homotopy Formula . 200 7.4.4 Applications of the Homotopy Formula . 201 7.5 Rectifiable Currents with Integer Multiplicity . 204 7.6Slicing...............................214 7.7 The Deformation Theorem . 221 7.8 Proof of the Unscaled Deformation Theorem . 228 7.9 Applications of the Deformation Theorem . 234 8 Currents and the Calculus of Variations 239 8.1 Proof of the Compactness Theorem . 239 8.1.1 Integer-Multiplicity 0-Currents . 240 8.1.2 A Rectifiability Criterion for Currents . 246 8.1.3 MBV Functions . 247 8.1.4 The Slicing Lemma . 253 8.1.5 The Density Lemma . 253 8.1.6 Completion of the Proof of the Compactness Theorem 256 8.2 The Flat Metric . 257 8.3 Existence of Currents Minimizing Variational Integrals . 261 8.3.1 Minimizing Mass . 261 8.3.2 Other Integrands and Integrals . 263 8.4 Density Estimates for Minimizing Currents . 269 9 Regularity of Mass-Minimizing Currents 275 9.1 Preliminaries . 276 9.2 The Height Bound and Lipschitz Approximation . 284 9.3 Currents defined by integrating over graphs . 292 9.4 Estimates for Harmonic Functions . 296 9.5 The Main Estimate . 310 9.6 The Regularity Theorem . 329 9.7Epilogue..............................334 Appendix 337 A.1 Transfinite Induction . 337 A.2 Dual Spaces . 339 A.2.1 The Dual of an Inner Product Space . 342 A.3 Line Integrals . 343 A.3.1 Exterior Differentiation . 344 iv CONTENTS A.4 Pullbacks and Exterior Derivatives . 346 Bibliography 351 Index of Notation 365 Preface Geometric measure theory has roots going back to ancient Greek mathe- matics. For considerations of the isoperimetric problem (to find 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 well- spring of questions in geometric measure theory. Named in honor of the nineteenth century Belgian physicist Joseph Plateau who studied surface tension phenomena in general, and soap films and soap bubbles in particu- lar, the question (in its original formulation) was to show that a fixed, simple closed curve in three-space will bound a surface of the type of a disc and hav- ing 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 surface to be a harmonic mapping (which one sees by studying the Dirichlet integral). For this effort he was awarded the Fields Medal in 1936. Unfortuately, Douglas’s methods do not adapt well to higher dimensions, so it is desirable to find other techniques with broader applicability. Enter the theory of currents. Currents are continuous linear functionals on spaces of differential forms. Brought to fruition by Federer and Fleming in the 1950s, currents turn out to be a natural language in which to formulate the sorts of extremal problems that arise in geometry. One can show that the natural differential operators in the subject are closed when acting on spaces of currents, and one can prove compactness and structure theorems for spaces of currents that satisfy certain natural bounds. These two facts are key to the study of generalized versions of the Plateau problem and related questions of geometric analysis. As a result, Federer and Fleming were able to prove the existence of a solution to the general Plateau problem in all dimensions and codimensions in 1960. v vi CONTENTS Today geometric measure theory, which is properly focused on the study of currents and their geometry, is a burgeoning field in its own right. Fur- thermore, the techniques of geometric measure theory are finding good use in complex geometry, in partial differential equations, and in many other parts of modern geometry. It is well to have a text that introduces the graduate student to key ideas in this subject. The present book is such a text. Demanding minimal background—only basic courses in calculus and linear algebra and real variables and measure theory—this book treats all the key ideas in the subject. These include the deformation theorem, the area and coarea formulas, the compactness theo- rem, the slicing theorem, and applications to fundamental questions about minimal surfaces that span given boundaries. In an effort to keep things as fundamental and near-the-surface as possible, we eschew generality and concentrate on the most essential results. As part of our effort to keep the exposition self-contained and accessible, we have limited our treatment of the regularity theory to proving almost-everywhere regularity of mass-minimizing hypersurfaces. We provide a full proof of the Lipschitz space estimate for harmonic functions that underlies the regularity of mass-minimizing hyper- surfaces. The notation in this subject—which is copious and complex—has been carefully considered by these authors and we have made strenuous effort to keep it as streamlined as possible. This is virtually the only graduate-level text in geometric measure theory that has figures and fully develops the subject; we feel that these figures add to the clarity of the exposition. It should also be stressed that this book provides considerable background to bring the student up to speed. This includes • measure theory • lower-dimensional measures and Carath´eodory’s construction • Haar measure • covering theorems and differentiation of measures • Poincar´einequalities • differential forms and Stokes’s theorem • a thorough introduction to distributions and currents CONTENTS vii Some students will find that they can skip certain of the introductory mate- rial; but it is useful to have it all present as a resource, and for reference. We have also made a special effort to keep this book self-contained. We do not want the reader running off to other sources for key ideas; he or she should be able to read this book while sitting at home. Geometric measure theory uses techniques from geometry, measure the- ory, analysis, and partial differential equations. This book showcases all these methodologies, and explains the ways in which they interact. The result is a rich symbiosis which is both rewarding and educational. The subject of geometric measure theory deserves to be known to a broad audience, and we hope that the present text will facilitate the dissemination to and appreciation of the subject for a new generation of mathematicians. It has been our pleasure to record these topics in a definitive and accessible and, we hope, lively form. We hope that the reader will derive the same satisfaction in studying these ideas in the present text. Of course we welcome comments and criticisms, so that the book may be kept lively and current and of course as accurate as possible. We are happy to thank Randi D. Ruden and Hypatia S. R. Krantz for genealogical help and Susan Parks for continued strength. It is a particular pleasure to thank our teachers and mentors, Frederick J. Almgren and Her- bert Federer, for their inspiration and for the model that they set.
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