This research-level monograph on harmonic maps between singular spaces sets out much new material on the theory, bringing all the research together for the first time in one place. Riemannian polyhedra are a class of such spaces that are especially suitable to serve as the domain of definition for harmonic maps. Their properties are considered in detail, with many examples being given, and potential theory on Riemmanian polyhedra is also considered. The work will serve as a concise source and reference for all researchers working in this field or a similar one.
Author(s): J. Eells, B. Fuglede, M. Gromov
Series: Cambridge Tracts in Mathematics
Edition: 1
Publisher: Cambridge University Press
Year: 2001
Language: English
Pages: 310
Contents......Page 6
Gromov's Preface......Page 10
Authors' Preface......Page 12
The smooth framework ......Page 14
Harmonic and Dirichlet spaces ......Page 17
Riemannian polyhedra ......Page 18
Harmonic functions on X ......Page 19
Geometric examples ......Page 20
Maps between polyhedra ......Page 21
Harmonic maps ......Page 24
Singular frameworks ......Page 25
Part I. Domains, targets, examples ......Page 27
Harmonic spaces ......Page 28
Dirichlet structures on a space ......Page 33
Geodesic spaces ......Page 37
Example 3.1. Riemannian manifolds ......Page 43
Example 3.3. Finsler structure on a manifold ......Page 44
Example 3.5. Lie algebras of vector fields on a manifold ......Page 46
Example 3.6. Riemannian Lipschitz manifolds ......Page 50
Example 3.7. The infinite dimensional torus T^infty ......Page 52
Lip continuous map. Lip homeomorphism ......Page 54
Simplicial complex ......Page 55
Polyhedron ......Page 57
Circuit ......Page 58
Lip polyhedron ......Page 59
Riemannian polyhedron ......Page 60
The intrinsic distance d_X ......Page 64
Local structure in terms of cubes ......Page 70
Uniform estimate of ball volumes ......Page 73
Part II. Potential theory on polyhedra ......Page 75
The Sobolev space W^{1,2}(X) ......Page 76
A Poincare inequality ......Page 81
Weakly harmonic and weakly sub/superharmonic functions ......Page 85
Unique continuation of harmonic functions ......Page 90
Proof of Theorem 6.1 in the locally bounded case ......Page 92
Completion of the proof of Theorem 6.1 ......Page 101
Holder continuity ......Page 104
Harmonic space structure ......Page 112
The Dirichlet space L_0{1,2}(X) ......Page 117
The Green kernel ......Page 121
Quasitopology and fine topology ......Page 138
Sobolev functions on quasiopen sets ......Page 140
Subharmonicity of convex functions ......Page 142
Example 8.1. 1-dimensional Riemannian polyhedra ......Page 143
Example 8.2. The need for dimensional homogeneity ......Page 144
Example 8.5. A kind of connected sum of polyhedra ......Page 145
Example 8.6. Riemannian joins of Riemannian manifolds ......Page 146
Example 8.8. Conical singular Riemannian spaces ......Page 147
Example 8.9. Normal analytic spaces with singularities ......Page 148
Example 8.10. The Kobayashi distance ......Page 151
Example 8.11. Riemannian branched coverings ......Page 152
Example 8.12. The quotient M/K ......Page 155
Example 8.13. Riemannian orbifolds ......Page 159
Example 8.14. Buildings of Bruhat-Tits ......Page 160
Part III. Maps between polyhedra ......Page 163
Energy density and energy ......Page 164
Energy of maps into Riemannian manifolds ......Page 175
Energy of maps into Riemannian polyhedra ......Page 186
The volume of a map ......Page 189
10. Holder continuity of energy minimizers ......Page 191
The case of a target of nonpositive curvature ......Page 192
Proof of Theorem 10.1 ......Page 202
The case of a target of upper bounded curvature ......Page 205
11. Existence of energy minimizers ......Page 211
The case of free homotopy ......Page 213
The Dirichlet problem relative to a homotopy class ......Page 219
The ordinary Dirichlet problem ......Page 221
The case of 2-dimensional manifold domains ......Page 224
Questions and remarks ......Page 226
A concept of harmonic map ......Page 230
Weakly harmonic maps into a Riemannian manifold ......Page 234
Holder continuity revisited ......Page 243
Totally geodesic maps ......Page 246
Geodesics as harmonic maps ......Page 249
Jensen's inequality for maps ......Page 254
Harmonic maps from a 1-dimensional Riemannian polyhedron ......Page 256
Harmonic morphisms between harmonic spaces ......Page 260
Harmonic morphisms between Riemannian polyhedra ......Page 262
Harmonic morphisms into Riemannian manifolds ......Page 264
Subpartitioning Lemma ......Page 272
Directional energies ......Page 274
Trace maps ......Page 275
Introduction and results ......Page 277
Embedding Y into an NPC cone ......Page 278
Holder continuity of the minimiser ......Page 281
Proof of Theorem 15.1 ......Page 286
Lipschitz continuity of the minimizer ......Page 288
Bibliography ......Page 290
Special symbols ......Page 304
Index ......Page 307
