Harmonic Maps, Conservation Laws and Moving Frames

Overview

This accessible introduction to harmonic map theory and its analytical aspects, covers recent developments in the regularity theory of weakly harmonic maps. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. It then presents a guided tour into the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions. A presentation of "exotic" functional ...
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Overview

This accessible introduction to harmonic map theory and its analytical aspects, covers recent developments in the regularity theory of weakly harmonic maps. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. It then presents a guided tour into the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions. A presentation of "exotic" functional spaces from the theory of harmonic analysis is given and these tools are then used for proving regularity results. The importance of conservation laws is stressed and the concept of a "Coulomb moving frame" is explained in detail. The book ends with further applications and illustrations of Coulomb moving frames to the theory of surfaces.
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Editorial Reviews

From The Critics
This book provides an accessible and self-contained introduction to harmonic map theory and its analytical aspects, covering recent developments in the regularity theory of weakly harmonic maps and stressing the interplay between geometry, the role of symmetrics, and weak solutions. The theory of completely integrable systems for harmonic maps is explained, and two chapters are devoted to recent results on the regularity of weak solutions. Hélein is affiliated with Ecole Normale Supérieure de Cachan. The first edition of the book was published in French as Applications harmoniques, lois de conservation et repeéres mobiles, Diderot Editeur, 1996. An English version, translated by L. Almeida, was published by Diderot Editeur in 1997. Annotation c. Book News, Inc., Portland, OR
From the Publisher
"...an excellent overview...The reviewer has really enjoyed reading this book." MATH

"The presentation of the material is excellent. The book is as self-contained as it can be....The concepts that are introduced are very carefully motivated. Because the book also gives an overview about recent research results (even updated for the second edition), it is not only a pleasure to read, but also a very good reference book for research." Mathematical Reviews

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Product Details

  • ISBN-13: 9781887933308
  • Publisher: Diderot Publisher, Arts & Science
  • Publication date: 11/1/1997
  • Series: New Essays Ser.
  • Pages: 308

Table of Contents

Foreword
Introduction
Acknowledgements
Notation
1 Geometric and analytic setting 1
1.1 The Laplacian on (M,g) 2
1.2 Harmonic maps between two Riemannian manifolds 5
1.3 Conservation laws for harmonic maps 11
1.4 Variational approach: Sobolev spaces 31
1.5 Regularity of weak solutions 46
2 Harmonic maps with symmetry 49
2.1 Backlund transformation 50
2.2 Harmonic maps with values into Lie groups 58
2.3 Harmonic maps with values into homogeneous spaces 82
2.4 Synthesis: relation between the different formulations 95
2.5 Compactness of weak solutions in the weak topology 101
2.6 Regularity of weak solutions 109
3 Compensations and exotic function spaces 114
3.1 Wente's inequality 115
3.2 Hardy spaces 128
3.3 Lorentz spaces 135
3.4 Back to Wente's inequality 145
3.5 Weakly stationary maps with values into a sphere 150
4 Harmonic maps without symmetry 165
4.1 Regularity of weakly harmonic maps of surfaces 166
4.2 Generalizations in dimension 2 187
4.3 Regularity results in arbitrary dimension 193
4.4 Conservation laws for harmonic maps without symmetry 205
5 Surfaces with mean curvature in L[superscript 2] 221
5.1 Local results 224
5.2 Global results 237
5.3 Willmore surfaces 242
5.4 Epilogue: Coulomb frames and conformal coordinates 244
References 254
Index 263
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