The Pullback Equation for Differential Forms

  • Gyula Csató
  • Bernard Dacorogna
  • Olivier Kneuss

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 83)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
    Pages 1-29
  3. Exterior and Differential Forms

    1. Front Matter
      Pages 31-31
    2. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 33-74
    3. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 75-90
    4. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 91-97
  4. Hodge–Morrey Decomposition and Poincaré Lemma

    1. Front Matter
      Pages 99-99
    2. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 101-120
    3. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 121-133
    4. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 135-146
    5. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 147-177
    6. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 179-188
  5. The Case k = n

    1. Front Matter
      Pages 189-189
    2. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 191-210
    3. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 211-252
  6. The Case 0 ≤ k ≤ n−1

    1. Front Matter
      Pages 253-253
    2. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 255-265
    3. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 267-283
    4. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 285-317
    5. Gyula Csató, Bernard Dacorogna, Olivier Kneuss
      Pages 319-331

About this book

Introduction

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.

Keywords

Hodge decomposition Hölder spaces Poincaré lemma equivalence of differential forms global Darboux theorem local Darboux theorem pullback equation

Authors and affiliations

  • Gyula Csató
    • 1
  • Bernard Dacorogna
    • 2
  • Olivier Kneuss
    • 3
  1. 1., Section de MathématiquesEcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2., Section de MathématiquesÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3., Section de MathématiquesEcole Polytechnique Fédérale de LausanneLausanneSwitzerland

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-8176-8313-9
  • Copyright Information Springer Science+Business Media, LLC 2012
  • Publisher Name Birkhäuser Boston
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-8176-8312-2
  • Online ISBN 978-0-8176-8313-9
  • About this book