Table Of Content

Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . 11 1.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Carathéodory’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 Carathéodory’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 Poincaré Inequalities 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 Poincaré 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 7.6 Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 particular, the question (in its original formulation) was to show that a fixed, simple closedcurve in three-space willbound a surface of the typeof a discand 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 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 ofcurrents,and onecanprovecompactnessand structuretheoremsforspaces ofcurrentsthatsatisfycertainnaturalbounds. Thesetwofactsarekeyto 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 techniquesof geometricmeasuretheory 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 theorem, 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-containedand accessible,wehavelimitedour treatmentof the regularitytheorytoprovingalmost-everywhereregularityofmass-minimizing hypersurfaces. We provide a full proof of the Lipschitz space estimate for harmonic functions that underlies the regularity of mass-minimizing hypersurfaces. 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. Itshouldalsobestressedthatthisbook providesconsiderablebackground to bring the student up to speed. This includes • measure theory • lower-dimensional measures and Carathéodory’s construction • Haar measure • covering theorems and differentiation of measures • Poincaré inequalities • 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 theory,analysis,and partialdifferentialequations. Thisbook showcases allthese methodologies, and explains the ways in which they interact. The result is a rich symbiosis which is both rewarding and educational. The subjectof geometricmeasure theory deservesto 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 instudying theseideas inthe presenttext. Of course wewelcome 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 theirinspiration and for themodel that they set. Geometric measure theory is a different subject because of their work. —Steven G. Krantz —Harold R. Parks Chapter 1 Basics Our purpose in this chapter will be to establish notation and terminology. The reader should already be acquainted with most of the concepts discussed and, thus might wish to skim the chapter or skip ahead, returning if clarifi- cation is needed. 1.1 Smooth Functions The real numbers will be denoted by R. In this book, we will be concerned with questions of geometric analysis in an N-dimensional Euclidean space. That is, we will work in the space RN of ordered N-tuples of real numbers. The inner product x·y of two elements x,y ∈ RN is defined by setting N x·y = Xx y , i i i=1 where x = (x ,x ,...,x ) and y = (y ,y ,...,y ). 1 2 N 1 2 N Of course, the inner product is a symmetric, bilinear, positive definite function on RN ×RN. The norm of the element x ∈ RN, denoted |x|, is defined by setting √ |x| = x·x, (1.1) aswemaysincetheright-handsideof (1.1)isalwaysnon-negative. The stan- dardorthonormal basis elements forRN willbedenotedbye , i = 1,2,...,N. i Specifically, e is the vector with N entries, all of which are 0s except the i 1 2 BASICS ith entry which is 1. For computational purposes, elements of RN should be considered column vectors. For typographical purposes, column vectors can waste space on the page, so we sometimes take the liberty of using row vector notation, as we did above. The open ball of radius r > 0 centered at x will be denoted B(x,r) and is defined by setting B(x,r) = { y ∈ RN : |x−y| < r }. The closed ball of radius r ≥ 0 centered at x will be denoted B(x,r) and is defined by setting B(x,r) = { y ∈ RN : |x−y| ≤ r }. The standard topology on the space RN is defined by letting the open sets consist of all arbitrary unions of open balls. The closed sets are then defined to be the complements of the open sets. For any subset A of RN (or of any topological space), there is a largest open set contained in A. That ˚ set, denoted A, is called the interior of A. Similarly, A is contained in a smallest closed set containing A and that set, denoted A, is calledthe closure of A. The topological boundary of A denoted ∂A is defined by setting ∂A= A\A˚. Remark 1.1.1 (1) At this juncture, the only notion of boundary in sight is that of the topological boundary. Since later we shall be led to define another notion of boundary, we are taking care to emphasize that the present definition is the topological one. When it is clear from context that we are discussing the topological boundary, then we will refer simply to the “boundary of A.” ˚ (2) The notations A and A for the interior and closure, respectively,of the set A are commonly used but are not universal. A variety of notations is used for the topological boundary of A, and ∂A is one of the more popular choices. Let U ⊆ RN be any open set. A function f : U → RM is said to be continuously differentiable of order k, or Ck, if f possesses all partial