| Description |
1 online resource (xxv, 264 pages) |
| Series |
Cambridge tracts in mathematics ; 150 |
|
Cambridge tracts in mathematics ; 150.
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| Contents |
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 -- 1.3.1 Symmetries on N 12 -- 1.3.2 Symmetries on M: the stress-energy tensor 18 -- 1.3.3 Consequences of theorem 1.3.6 24 -- 1.4 Variational approach: Sobolev spaces 31 -- 1.4.1 Weakly harmonic maps 37 -- 1.4.2 Weakly Noether harmonic maps 42 -- 1.4.3 Minimizing maps 42 -- 1.4.4 Weakly stationary maps 43 -- 1.4.5 Relation between these different definitions 43 -- 1.5 Regularity of weak solutions 46 -- 2 Harmonic maps with symmetry 49 -- 2.1 Backlund transformation 50 -- 2.1.1 S[superscript 2]-valued maps 50 -- 2.1.2 Maps taking values in a sphere S[superscript n], n [greater than or equal] 2 54 -- 2.1.3 Comparison 56 -- 2.2 Harmonic maps with values into Lie groups 58 -- 2.2.1 Families of curvature-free connections 65 -- 2.2.2 The dressing 72 -- 2.2.3 Uhlenbeck factorization for maps with values in U(n) 77 -- 2.2.4 S[superscript 1]-action 79 -- 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 -- 3.1.1 The inequality on a plane domain 115 -- 3.1.2 The inequality on a Riemann surface 119 -- 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 -- 4.4.1 Conservation laws 206 -- 4.4.2 Isometric embedding of vector-bundle-valued differential forms 211 -- 4.4.3 A variational formulation for the case m = n = 2 and p = 1 215 -- 4.4.4 Hidden symmetries for harmonic maps on surfaces? 218 -- 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 |
| Summary |
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 |
| Bibliography |
Includes bibliographical references (pages 254-262) and index |
| Notes |
Print version record |
| Subject |
Harmonic maps.
|
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Riemannian manifolds.
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MATHEMATICS -- Topology.
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Harmonic maps
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Riemannian manifolds
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Erhaltungssatz
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Harmonische Abbildung
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| Form |
Electronic book
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| ISBN |
0511016603 |
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9780511016608 |
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9780511543036 |
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0511543034 |
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