Geometric integration theory

381 Pages·2008·1.249 MB·English

by  Steven G. Krantz, Harold R. Parks

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 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´e 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 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
7.6 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 particu-
lar, 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 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
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 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-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 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.
Itshouldalsobestressedthatthisbook providesconsiderablebackground
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´e 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 the-
ory,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 func-
tion 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