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Harmonic Maps, Conservation Laws and Moving Frames

ISBN-10: 0521811600

ISBN-13: 9780521811606

Edition: 2nd 2002 (Revised)

Authors: B. Bollobas, W. Fulton, Fr�d�ric H�lein, A. Katok, F. Kirwan

List price: $154.99
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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.
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Book details

List price: $154.99
Edition: 2nd
Copyright year: 2002
Publisher: Cambridge University Press
Publication date: 6/13/2002
Binding: Hardcover
Pages: 292
Size: 6.25" wide x 9.00" long x 0.75" tall
Weight: 1.386
Language: English

Geometric and analytic setting
The Laplacian on (M, g)
Harmonic maps between two Riemannian manifolds
Conservation laws for harmonic maps
Symmetries on N
Symmetries on M: the stress-energy tensor
Consequences of theorem 1.3.6
Variational approach: Sobolev spaces
Weakly harmonic maps
Weakly Noether harmonic maps
Minimizing maps
Weakly stationary maps
Relation between these different definitions
Regularity of weak solutions
Harmonic maps with symmetry
Backlund transformation
S[superscript 2]-valued maps
Maps taking values in a sphere S[superscript n], n [greater than or equal] 2
Harmonic maps with values into Lie groups
Families of curvature-free connections
The dressing
Uhlenbeck factorization for maps with values in U(n)
S[superscript 1]-action
Harmonic maps with values into homogeneous spaces
Synthesis: relation between the different formulations
Compactness of weak solutions in the weak topology
Regularity of weak solutions
Compensations and exotic function spaces
Wente's inequality
The inequality on a plane domain
The inequality on a Riemann surface
Hardy spaces
Lorentz spaces
Back to Wente's inequality
Weakly stationary maps with values into a sphere
Harmonic maps without symmetry
Regularity of weakly harmonic maps of surfaces
Generalizations in dimension 2
Regularity results in arbitrary dimension
Conservation laws for harmonic maps without symmetry
Conservation laws
Isometric embedding of vector-bundle-valued differential forms
A variational formulation for the case m = n = 2 and p = 1
Hidden symmetries for harmonic maps on surfaces?
Surfaces with mean curvature in L[superscript 2]
Local results
Global results
Willmore surfaces
Epilogue: Coulomb frames and conformal coordinates