The Pullback Equation for Differential Forms

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Preface

In the present book we study the pullback equation for differential forms
ϕ∗(g) = f ,
namely, given two differential k-forms f and g we want to discuss the equivalence of
such forms. This turns out to be a system of nonlinear first-order partial differential
equations in the unknown map ϕ.
The problem that we study here is a particular case of the equivalence of tensors
which has received considerable attention. However, the pullback equation for differential
forms has quite different features than those for symmetric tensors, such
as Riemannian metrics, which has also been studied a great deal. In more physical
terms, the problem of equivalence of forms can also be seen as a problem of mass
transportation.
This is an important problem in geometry and in analysis. It has been extensively
studied, in the cases k =2 and k =n, but much less when 3≤k ≤n−1. The problem
considered here of finding normal forms (Darboux theorem, Pfaff normal form) is
a fundamental question in symplectic and contact geometry. With respect to the
literature in geometry, the main emphasis of the book is on regularity and boundary
conditions. Indeed, special attention has been given to getting optimal regularity;
this is a particularly delicate point and requires estimates for elliptic equations and
fine properties of H¨older spaces.
In the case of volume forms (i.e., k = n), our problem is clearly related to the
widely studied subject of optimal mass transportation. However, our analysis is not
in this framework. As stated before, the two main points of our analysis are that we
provide optimal regularity in H¨older spaces and, at the same time, we are able to
handle boundary conditions.
Our book will hopefully appeal to both geometers and analysts. In order to make
the subject more easily attractive for the analysts, we have reduced as much as possible
the notations of geometry. For example, we have restricted our attention to
domains in Rn, but it goes without saying that all results generalize to manifolds
with or without boundary.
In Part I we gather some basic facts about exterior and differential forms that are
used throughout Parts II and IV. Most of the results are standard, but they are presented
so that the reader may be able to grasp the main results of the subject without
getting too involved with the terminology and concepts of differential geometry.
Part II presents the classical Hodge decomposition following the proof of Morrey,
but with some variants, notably in our way of deriving the Gaffney inequality. We
also give applications to several versions of the Poincar´e lemma that are constantly
used in the other parts of the book. Part II can be of interest independently of the
main subject of the book.
Part III discusses the case k = n. We have tried in this part to make it, as much
as possible, independent of the machinery of differential forms. Indeed, Part III can
essentially be read with no reference to the other parts of the work, except for the
properties of H¨older spaces presented in Part V.
Part IV deals with the general case. Emphasis in this part is given to the symplectic
case k = 2. We also briefly deal with the simpler cases k = 0,1, n−1. The
case 3 ≤ k ≤ n−2 is much harder and we are able to obtain results only for forms
having a special structure. The difficulty is already at the algebraic level.
In Part V we gather several basic properties of H¨older spaces that are used extensively
throughout the book. Due to the nonlinearity of the pullback equation, H¨older
spaces are much better adapted than Sobolev spaces. The literature on H¨older spaces
is considerably smaller than the one on Sobolev spaces. Moreover, the results presented
here cannot be found solely in a single reference. We hope that this part will
be useful to mathematicians well beyond those who are primarily interested in the
pullback equation.
Acknowledgments Several results of Part IV find their origins from joint works
and discussions with S. Bandyopadhyay. During the preparation of the manuscript,
we have benefited from many helpful comments by P. Bousquet, G. Cupini, W.
Gangbo, N. Kamran, T. Ratiu, K.D. Semmler, D. Serre and D. Ye. The discussions
with M. Troyanov have been particularly fruitful. We also thank H. Br´ezis for accepting,
with enthusiasm, our book in the Birkh¨auser series that he edits.
The research of the third author has been, in part, subsidized by a grant of the
Fonds National Suisse de la Recherche Scientifique.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Exterior and Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Definitions and Basic Properties of Exterior Forms . . . . . . . . 3
1.2.2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Hodge–Morrey Decomposition and Poincar´e Lemma . . . . . . . . . . . . 10
1.3.1 A General Identity and Gaffney Inequality . . . . . . . . . . . . . . . 10
1.3.2 The Hodge–Morrey Decomposition . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 First-Order Systems of Cauchy–Riemann Type . . . . . . . . . . . 12
1.3.4 Poincar´e Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 The Case of Volume Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.2 The One-Dimensional Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.3 The Case f · g > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.4 The Case with No Sign Hypothesis on f . . . . . . . . . . . . . . . . . 19
1.5 The Case 0 ≤ k ≤ n−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 The Flow Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.2 The Cases k = 0 and k = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5.3 The Case k = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.4 The Case 3 ≤ k ≤ n−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 H¨older Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6.1 Definition and Extension of H¨older Functions . . . . . . . . . . . . 25
1.6.2 Interpolation, Product, Composition and Inverse . . . . . . . . . . 27
1.6.3 Smoothing Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Part I Exterior and Differential Forms
2 Exterior Forms and the Notion of Divisibility . . . . . . . . . . . . . . . . . . . . . . 33
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.1 Exterior Forms and Exterior Product . . . . . . . . . . . . . . . . . . . . 34
2.1.2 Scalar Product, Hodge Star Operator and Interior Product . . 36
2.1.3 Pullback and Dimension Reduction . . . . . . . . . . . . . . . . . . . . . 39
2.1.4 Canonical Forms for 1, 2, (n−2) and (n−1)-Forms . . . . . . . 42
2.2 Annihilators, Rank and Corank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.1 Exterior and Interior Annihilators . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.2 Rank and Corank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.3 Properties of the Rank of Order 1 . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.3.1 Definition and First Properties . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.3.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.3.3 Some More Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.3.4 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 Tangential and Normal Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3 Gauss–Green Theorem and Integration-by-Parts Formula . . . . . . . . . 87
4 Dimension Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1 Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Reduction Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Part II Hodge–Morrey Decomposition and Poincar´e Lemma
5 An Identity Involving Exterior Derivatives and Gaffney Inequality . . 101
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 An Identity Involving Exterior Derivatives . . . . . . . . . . . . . . . . . . . . . . 103
5.2.1 Preliminary Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.2 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3 Gaffney Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3.1 An Elementary Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3.2 A Generalization of the Boundary Condition . . . . . . . . . . . . . 115
5.3.3 Gaffney-Type Inequalities in Lp and H¨older Spaces . . . . . . . . 118
6 The Hodge–Morrey Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1 Properties of Harmonic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Existence of Minimizers and Euler–Lagrange Equation . . . . . . . . . . . 124
6.3 The Hodge–Morrey Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.4 Higher Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 First-Order Elliptic Systems of Cauchy–Riemann Type . . . . . . . . . . . . . 135
7.1 System with Prescribed Tangential Component . . . . . . . . . . . . . . . . . . 135
7.2 System with Prescribed Normal Component . . . . . . . . . . . . . . . . . . . . 140
7.3 Weak Formulation for Closed Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.4 Equivalence Between Hodge Decomposition and Cauchy–
Riemann-Type Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8 Poincar´e Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.1 The Classical Poincar´e Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.2 Global Poincar´e Lemma with Optimal Regularity . . . . . . . . . . . . . . . . 148
8.3 Some Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.4 Poincar´e Lemma with Dirichlet Boundary Data . . . . . . . . . . . . . . . . . 157
8.5 Poincar´e Lemma with Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.5.1 A First Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.5.2 A Second Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.5.3 Some Technical Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9 The Equation divu = f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.1 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.2 Regularity of Divergence-Free Vector Fields . . . . . . . . . . . . . . . . . . . . 181
9.3 Some More Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.3.1 A First Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.3.2 A Second Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Part III The Case k = n
10 The Case f · g > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.1 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.2 The Flow Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.3 The Fixed Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.4 Two Proofs of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
10.4.1 First Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
10.4.2 Second Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
10.5 A Constructive Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
11 The CaseWithout Sign Hypothesis on f . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.2 Remarks and Related Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
11.3 Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
11.4 Radial Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
11.5 Concentration of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
11.6 Positive Radial Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Part IV The Case 0 ≤ k ≤ n−1
12 General Considerations on the Flow Method . . . . . . . . . . . . . . . . . . . . . . 255
12.1 Basic Properties of the Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
12.2 A Regularity Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
12.3 The Flow Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
13 The Cases k = 0 and k = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
13.1 The Case of 0-Forms and of Closed 1-Forms . . . . . . . . . . . . . . . . . . . . 267
13.1.1 The Case of 0-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
13.1.2 The Case of Closed 1-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
13.2 Darboux Theorem for 1-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
13.2.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
13.2.2 A Technical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
14 The Case k = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
14.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
14.2 Local Result for Forms with Maximal Rank . . . . . . . . . . . . . . . . . . . . 286
14.3 Local Result for Forms of Nonmaximal Rank . . . . . . . . . . . . . . . . . . . 290
14.3.1 The Theorem and a First Proof . . . . . . . . . . . . . . . . . . . . . . . . . 290
14.3.2 A Second Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
14.4 Global Result with Dirichlet Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
14.4.1 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
14.4.2 The Flow Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
14.4.3 The Key Estimate for Regularity . . . . . . . . . . . . . . . . . . . . . . . 295
14.4.4 The Fixed Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
14.4.5 A First Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . 308
14.4.6 A Second Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . 314
15 The Case 3 ≤ k ≤ n−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
15.1 A General Theorem for Forms of Rank = k . . . . . . . . . . . . . . . . . . . . . 319
15.2 The Case of (n−1)-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
15.2.1 The Case of Closed (n−1)-Forms . . . . . . . . . . . . . . . . . . . . . . 321
15.2.2 The Case of Nonclosed (n−1)-Forms . . . . . . . . . . . . . . . . . . . 322
15.3 Simultaneous Resolutions and Applications . . . . . . . . . . . . . . . . . . . . 324
15.3.1 Simultaneous Resolution for 1-Forms . . . . . . . . . . . . . . . . . . . 324
15.3.2 Applications to k-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
Part V H¨older Spaces
16 H¨older Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
16.1 Definitions of Continuous and H¨older Continuous Functions . . . . . . 335
16.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
16.1.2 Regularity of Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
16.1.3 Some Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
16.2 Extension of Continuous and H¨older Continuous Functions . . . . . . . 341
16.2.1 The Main Result and Some Corollaries . . . . . . . . . . . . . . . . . . 341
16.2.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
16.2.3 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
16.3 Compact Imbeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
16.4 A Lower Semicontinuity Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
16.5 Interpolation and Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
16.5.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
16.5.2 Product and Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
16.6 Composition and Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
16.6.1 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
16.6.2 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
16.6.3 A Further Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
16.7 Difference of Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
16.7.1 A First Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
16.7.2 A Second Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
16.7.3 A Third Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
16.8 The Smoothing Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
16.8.1 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
16.8.2 A First Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
16.8.3 A Second Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
16.9 Smoothing Operator for Differential Forms . . . . . . . . . . . . . . . . . . . . . 396
Part VI Appendix
17 Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
18 An Abstract Fixed Point Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
19 Degree Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
19.1 Definition and Main Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
19.2 General Change of Variables Formula . . . . . . . . . . . . . . . . . . . . . . . . . 418
19.3 Local and Global Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435