Harmonic Maps, Conservation Laws and Moving Frames

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.

ISBN 0521811600   Volumes 1 ISBN13 9780521811606 (What's this?)   Weight (grammes) 627 Publisher Cambridge University Press   Published in Cambridge Imprint Cambridge University Press   Series editor Bollobas, B., Fulton, W., Katok, A. Format Hardback   Series ISSN 150 Publication date 13 Jun 2002   Series title Cambridge Tracts in Mathematics Library of Congress QA614.73 .H45 2002   Height (mm) 228 DEWEY 514.74   Width (mm) 152 DEWEY edition DC21   Spine width (mm) 22 Pages 290   Academic level Professional / Scholarly

		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
		More...

'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

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