Harmonic morphisms between Riemannian manifolds

Paul Baird and John C. Wood

London Mathematical Society Monographs, No. 29,
Oxford University Press
(2003)

Description of the book
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Contents

Introduction xi

Part I Basic Facts on Harmonic Morphisms 17

1 Complex-valued harmonic morphisms on three-dimensional Euclidean space 3
1.1 Definition and characterization 3
1.2 Generating harmonic morphisms 6
1.3 A converse 9
1.4 Direction and displacement maps 14
1.5 Examples 17
1.6 A global theorem 21
1.7 Notes and comments 23
2 Riemannian manifolds and conformality 25
2.1 Riemannian manifolds 25
2.2 The Laplacian on a Riemannian manifold 35
2.3 Weakly conformal maps 40
2.4 Horizontally weakly conformal maps 45
2.5 Conformal foliations 54
2.6 Notes and comments 62
3 Harmonic mappings between Riemannian manifolds 65
3.1 Calculus on vector bundles 65
3.2 Second fundamental form and tension field 69
3.3 Harmonic mappings 71
3.4 The stress-energy tensor 81
3.5 Minimal branched immersions 84
3.6 Second variation of the energy and stability 91
3.7 Volume and energy 94
3.8 Notes and comments 100
4 Fundamental properties of harmonic morphisms 106
4.1 The Definition 106
4.2 Characterization 108
4.3 General properties 111
4.4 The symbol 114
4.5 The mean curvature of the fibres 118
4.6 Further consequences of the fundamental equations 124
4.7 Foliations which produce harmonic morphisms 128
4.8 Second variation 132
4.9 Notes and comments 136
5 Harmonic morphisms defined by polynomials 141
5.1 Entire harmonic morphisms between Euclidean spaces 141
5.2 Horizontally conformal polynomial maps 143
5.3 Orthogonal multiplications 148
5.4 Clifford systems 151
5.5 Quadratic harmonic morphisms 156
5.6 Homogeneous polynomial maps 162
5.7 Applications to horizontally weakly conformal maps 167
5.8 Notes and comments 169

Part II Twistor Methods 173

6 Mini-twistor theory on three-dimensional space forms 175
6.1 Factorization of harmonic morphisms from 3-manifolds 175
6.2 Geodesics on a three-dimensional space form 180
6.3 The space of oriented geodesics on Euclidean 3-space 183
6.4 The space of oriented geodesics on the 3-sphere 185
6.5 The space of oriented geodesics on hyperbolic 3-space 188
6.6 Harmonic morphisms from three-dimensional space forms 189
6.7 Entire harmonic morphisms on space forms 194
6.8 Higher dimensions 199
6.9 Notes and comments 203
7 Twistor methods 206
7.1 The twistor space of a Riemannian manifold 206
7.2 Kählerian twistor spaces 211
7.3 The twistor space of the 4-sphere 214
7.4 The twistor space of Euclidean 4-space 216
7.5 The twistor spaces of complex projective 2-space 217
7.6 The twistor space of an anti-self-dual 4-manifold 219
7.7 Adapted Hermitian structures 220
7.8 Superminimal surfaces 223
7.9 Hermitian structures from harmonic morphisms 228
7.10 Harmonic morphisms from Hermitian structures 231
7.11 Harmonic morphisms from Euclidean 4-space 236
7.12 Harmonic morphisms from the 4-sphere 239
7.13 Harmonic morphisms from complex projective 2-space 241
7.14 Harmonic morphisms from other Einstein 4-manifolds 243
7.15 Notes and comments 244
8 Holomorphic harmonic morphisms 250
8.1 Harmonic morphisms between almost Hermitian manifolds 250
8.2 Composition laws 254
8.3 Hermitian structures on open subsets of Euclidean spaces 257
8.4 The Weierstrass formulae 259
8.5 Reduction to odd dimensions and to spheres 262
8.6 General holomorphic harmonic morphisms on Euclidean spaces 266
8.7 Notes and comments 270
9 Multivalued harmonic morphisms 273
9.1 Multivalued mappings 274
9.2 Multivalued harmonic morphisms 276
9.3 Classes of Examples 281
9.4 An alternative treatment for space forms 283
9.5 Some specific examples 284
9.6 Behaviour on the branching set 288
9.7 Notes and comments 292

Part III Topological and Curvature Considerations 293

10 Harmonic morphisms from compact 3-manifolds 295
10.1 Seifert fibre spaces 295
10.2 Three-dimensional geometries 300
10.3 Harmonic morphisms and Seifert fibre spaces 302
10.4 Examples 305
10.5 Characterization of the metric 307
10.6 Propagation of fundamental quantities along the fibres 312
10.7 Notes and comments 317
11 Curvature considerations 319
11.1 The fundamental tensors 319
11.2 Curvature for a horizontally conformal submersion 320
11.3 Walczak's formula 327
11.4 Conformal maps between equidimensional manifolds 330
11.5 Curvature and harmonic morphisms 332
11.6 Weitzenböck formulae 338
11.7 Curvature for one-dimensional fibres 341
11.8 Entire harmonic morphisms on Euclidean space with totally geodesic fibres 347
11.9 Notes and comments 349
12 Harmonic morphisms with one-dimensional fibres 352
12.1 Topological restrictions 352
12.2 The normal form of the metric 360
12.3 Harmonic morphisms of Killing type 364
12.4 Harmonic morphisms of warped product type 366
12.5 Harmonic morphisms of type (T) 371
12.6 Uniqueness of types 374
12.7 Einstein manifolds 375
12.8 Harmonic morphisms from an Einstein 4-manifold 378
12.9 Constant curvature manifolds 383
12.10 Notes and comments 389
13 Reduction techniques 392
13.1 Isoparametric mappings 392
13.2 Eigen-harmonic morphisms 398
13.3 Reduction 399
13.4 Conformal changes of the metrics 402
13.5 Reduction to an ordinary differential equation 405
13.6 Reduction to a partial differential equation 413
13.7 Notes and comments 419

Part IV Further Developments 425

14 Harmonic morphisms between semi-Riemannian manifolds 427
14.1 Semi-Riemannian manifolds 427
14.2 Harmonic maps between semi-Riemannian manifolds 435
14.3 Harmonic maps between Lorentzian surfaces 438
14.4 Weakly conformal maps and stress-energy 440
14.5 Horizontally weakly conformal maps 444
14.6 Harmonic morphisms between semi-Riemannian manifolds 446
14.7 Harmonic morphisms between Lorentzian surfaces 449
14.8 Notes and comments 452
Appendix 456
A.1 Analytic aspects of harmonic functions 456
A.2 A regularity result for an equation of Yamabe type 460
A.3 A technical result on the symbol 462
A.4 Notes and comments 465
References 467
Glossary of notation 499
Index 502