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## Über dieses Buch

An important question in geometry and analysis is to know when two k-forms f and g are equivalent through a change of variables. The problem is therefore to find a map φ so that it satisfies the pullback equation: φ*(g) = f.

In more physical terms, the question under consideration can be seen as a problem of mass transportation. The problem has received considerable attention in the cases k = 2 and k = n, but much less when 3 ≤ k ≤ n–1. The present monograph provides the first comprehensive study of the equation.

The work begins by recounting various properties of exterior forms and differential forms that prove useful throughout the book. From there it goes on to present the classical Hodge–Morrey decomposition and to give several versions of the Poincaré lemma. The core of the book discusses the case k = n, and then the case 1≤ k ≤ n–1 with special attention on the case k = 2, which is fundamental in symplectic geometry. Special emphasis is given to optimal regularity, global results and boundary data. The last part of the work discusses Hölder spaces in detail; all the results presented here are essentially classical, but cannot be found in a single book. This section may serve as a reference on Hölder spaces and therefore will be useful to mathematicians well beyond those who are only interested in the pullback equation.

The Pullback Equation for Differential Forms is a self-contained and concise monograph intended for both geometers and analysts. The book may serve as a valuable reference for researchers or a supplemental text for graduate courses or seminars.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
The aim of this book is the study of the pullback equation ?* (g) = f. (1.1) More precisely, we want to find a map ? : Rn ?Rn; preferably we want this map to be a diffeomorphism that satisfies the above equation, where f and g are differential k-forms, 0 = k = n. Most of the time we will require these two forms to be closed.
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 2. Exterior Forms and the Notion of Divisibility

Abstract
The present chapter is divided into three parts. In Section 2.1, we recall the definitions and basic properties of exterior forms. All notions introduced there are standard and, therefore, our presentation will be very brief. We refer for further developments to the classic books on the subject–for example, Bourbaki [15], Bryant, Chern, Gardner, Goldschmidt and Griffiths [18], Godbillon [51], Godement [52], Greub [54], or Lang [67]. In what follows we will only consider the finite vector space Rn, n = 1, over R. However, we can obviously replace Rn by any finite n-dimensional vector space over a field K of characteristic 0.
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 3. Differential Forms

Abstract
In this section we recall the definitions and basic properties of differential forms on ℝn. Our presentation is very brief and for a detailed introduction on differential forms, we refer, for instance, to Abraham, Marsden and Ratiu [1], do Carmo [37], Lee [68], or Spivak [91].
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 4. Dimension Reduction

Abstract
We turn our attention to a very useful result, which is well known in the case of 2-forms. However, it can be extended in a straightforward way to the case of k-forms; it seems, however, that this extension has never been noticed. We will provide two proofs of the theorem. The first one is based on the Frobenius theorem and the second one is much more elementary and self-contained.
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 5. An Identity Involving Exterior Derivatives and Gaffney Inequality

Abstract
The aim of this chapter is twofold.
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 6. The Hodge–Morrey Decomposition

Abstract
We recall the definition of harmonic fields and of contractible sets. Let 0 ≤ k ≤ n be an integer.
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 7. First-Order Elliptic Systems of Cauchy–Riemann Type

Abstract
We first deal with the following boundary value problem:
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 8. Poincaré Lemma

Abstract
Our first result is the classical Poincaré lemma. Its proof is elementary and does not use the Hodge–Morrey decomposition. Its drawback (compare with Theorem 8.3) is that it does not provide the expected gain in regularity and is restricted to contractible sets.
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 9. The Equation divu = f

Abstract
which is constantly used in Chapter 10. Of course, most of the results can be found in Chapter 8. However, the proofs are much more elementary in this case and, in most cases, do not require the sophisticated machinery of Hodge–Morrey decomposition. They use only standard properties of the Laplacian. Therefore, for the convenience of the reader, we have gathered and proved the results in the present chapter.
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 10. The Case f · g > 0

Abstract
The main theorem of this chapter has been established by Dacorogna and Moser [33].
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 11. The Case Without Sign Hypothesis on f

Abstract
The aim of this chapter is to solve the problem
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 12. General Considerations on the Flow Method

Abstract
Let $$T > 0, \Omega \subset \mathbb{R}^{n} be an open set and g:[O,T] \times \overline{\Omega}\rightarrow \mathbb{R}^{N}.$$ Throughout the present chapter, when dealing with such maps, we write, depending on the context
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 13. The Cases k = 0 and k = 1

Abstract
We start with 0-forms. We begin our study with a local existence theorem.
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 14. The Case k = 2

Abstract
We recall, from Chapter 2, some notations that we will use throughout the present chapter. As usual, when necessary, we identify in a natural way 1-forms with vectors in ℝn.
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 15. The Case 3 ≤ k ≤ n−1

Abstract
The results that will be discussed in this chapter are strongly based on Bandyopadhyay, Dacorogna and Kneuss [9]. For related results see Turiel [97–102].
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 16. Hölder Continuous Functions

Abstract
We recall here the basic definitions of Hölder spaces. We use the following as references in the present chapter: Adams [2], Dacorogna [29], de la Llave and Obaya [36], Edmunds and Evans [40], Fefferman [42], Gilbarg and Trudinger [49] and Hörmander [55].
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 17. Necessary Conditions

Abstract
In the following proposition we gather some elementary necessary conditions (cf. [8], [9] and [31]).
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 18. An Abstract Fixed Point Theorem

Abstract
The following theorem is particularly useful when dealing with nonlinear problems, once good estimates are known for the linearized problem. We give it under a general form, because we have used it this way in Theorems 14.1 and 14.10. However, in many instances, Corollary 18.2 is amply sufficient. Our theorem will lean on the following hypotheses.
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Chapter 19. Degree Theory

Abstract
We begin recalling some results on the topological degree (see, e.g., [43] or [88] for further details). We start by defining the degree for C1 maps
Gyula Csató, Bernard Dacorogna, Olivier Kneuss

### Backmatter

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