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Harmonic Maps, Conservation Laws and Moving Frames : Cambridge Tracts in Mathematics - Frederic Helein

Harmonic Maps, Conservation Laws and Moving Frames

Cambridge Tracts in Mathematics


Published: 30th April 2013
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The author presents 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. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. The reader is then presented with 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 self-contained 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.

'The book is very well written and it contains truly beautiful geometrical analysis. It also contains a quick, direct introduction to the current research.' EMS Newsletter

Forewordp. ix
Introductionp. xiii
Acknowledgementsp. xxii
Notationp. xxiii
Geometric and analytic settingp. 1
The Laplacian on (M, g)p. 2
Harmonic maps between two Riemannian manifoldsp. 5
Conservation laws for harmonic mapsp. 11
Symmetries on Np. 12
Symmetries on M: the stress-energy tensorp. 18
Consequences of theorem 1.3.6p. 24
Variational approach: Sobolev spacesp. 31
Weakly harmonic mapsp. 37
Weakly Noether harmonic mapsp. 42
Minimizing mapsp. 42
Weakly stationary mapsp. 43
Relation between these different definitionsp. 43
Regularity of weak solutionsp. 46
Harmonic maps with symmetryp. 49
Backlund transformationp. 50
S[superscript 2]-valued mapsp. 50
Maps taking values in a sphere S[superscript n], n [greater than or equal] 2p. 54
Comparisonp. 56
Harmonic maps with values into Lie groupsp. 58
Families of curvature-free connectionsp. 65
The dressingp. 72
Uhlenbeck factorization for maps with values in U(n)p. 77
S[superscript 1]-actionp. 79
Harmonic maps with values into homogeneous spacesp. 82
Synthesis: relation between the different formulationsp. 95
Compactness of weak solutions in the weak topologyp. 101
Regularity of weak solutionsp. 109
Compensations and exotic function spacesp. 114
Wente's inequalityp. 115
The inequality on a plane domainp. 115
The inequality on a Riemann surfacep. 119
Hardy spacesp. 128
Lorentz spacesp. 135
Back to Wente's inequalityp. 145
Weakly stationary maps with values into a spherep. 150
Harmonic maps without symmetryp. 165
Regularity of weakly harmonic maps of surfacesp. 166
Generalizations in dimension 2p. 187
Regularity results in arbitrary dimensionp. 193
Conservation laws for harmonic maps without symmetryp. 205
Conservation lawsp. 206
Isometric embedding of vector-bundle-valued differential formsp. 211
A variational formulation for the case m = n = 2 and p = 1p. 215
Hidden symmetries for harmonic maps on surfaces?p. 218
Surfaces with mean curvature in L[superscript 2]p. 221
Local resultsp. 224
Global resultsp. 237
Willmore surfacesp. 242
Epilogue: Coulomb frames and conformal coordinatesp. 244
Referencesp. 254
Indexp. 263
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521811606
ISBN-10: 0521811600
Series: Cambridge Tracts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 292
Published: 30th April 2013
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 23.1 x 15.5  x 2.0
Weight (kg): 0.61
Edition Number: 2
Edition Type: Revised