Harmonic maps between Riemannian polyhedra:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2001
|
| Edition: | 1. publ. |
| Series: | Cambridge tracts in mathematics
142 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 296 S. |
| ISBN: | 0521773113 |
Staff View
MARC
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| 245 | 1 | 0 | |a Harmonic maps between Riemannian polyhedra |c J. Eells ; B. Fuglede |
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| 264 | 1 | |a Cambridge |b Cambridge Univ. Press |c 2001 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Gromov s Preface ix
Authors Preface xi
1. Introduction 1
The smooth framework 1
Harmonic and Dirichlet spaces 4
Riemannian polyhedra 5
Harmonic functions on X 6
Geometric examples 7
Maps between polyhedra 8
Harmonic maps 11
Harmonic morphisms 12
Singular frameworks 12
Part I. Domains, targets, examples 14
2. Harmonic spaces, Dirichlet spaces, and geodesic spaces 15
Harmonic spaces 15
Dirichlet structures on a space 20
Geodesic spaces 24
3. Examples of domains and targets 30
Example 3.1. Riemannian manifolds 30
Example 3.2. Almost Riemannian spaces 31
Example 3.3. Finsler structure on a manifold 31
Example 3.4. Metric associated to a holomorphic quadratic
differential 33
Example 3.5. Lie algebras of vector fields on a manifold 33
Example 3.6. Riemannian Lipschitz manifolds 37
Example 3.7. The infinite dimensional torus T°° 39
4. Riemannian polyhedra 41
Lip continuous map. Lip homeomorphism 41
Simplicial complex 42
Polyhedron 44
Circuit 45
vi Contents
Lip polyhedron 46
Riemannian polyhedron 47
The intrinsic distance dx 51
Local structure in terms of cubes 57
Uniform estimate of ball volumes 60
Part II. Potential theory on polyhedra 62
5. The Sobolev space W1 2(X). Weakly harmonic functions 63
The Sobolev space WX 2{X) 63
A Poincare inequality 68
Weakly harmonic and weakly sub/superharmonic functions 72
Unique continuation of harmonic functions 77
6. Harnack inequality and Holder continuity for weakly
harmonic functions 79
Proof of Theorem 6.1 in the locally bounded case 79
Completion of the proof of Theorem 6.1 88
Holder continuity 91
7. Potential theory on Riemannian polyhedra 99
Harmonic space structure 99
The Dirichlet space Ll 2{X) 104
The Green kernel 108
Quasitopology and fine topology 125
Sobolev functions on quasiopen sets 127
Subharmonicity of convex functions 129
8. Examples of Riemannian polyhedra and related spaces 130
Example 8.1. 1 dimensional Riemannian polyhedra 130
Example 8.2. The need for dimensional homogeneity 131
Example 8.3. The need for local chainability 132
Example 8.4. Manifolds as polyhedra 132
Example 8.5. A kind of connected sum of polyhedra 132
Example 8.6. Riemannian joins of Riemannian manifolds 133
Example 8.7. Riemannian orbit spaces 134
Example 8.8. Conical singular Riemannian spaces 134
Example 8.9. Normal analytic spaces with singularities 135
Example 8.10. The Kobayashi distance 138
Example 8.11. Riemannian branched coverings 139
Example 8.12. The quotient M/K 142
Example 8.13. Riemannian orbifolds 146
Example 8.14. Buildings of Bruhat Tits 147
Contents vii
Part III. Maps between polyhedra 150
9. Energy of maps 151
Energy density and energy 151
Energy of maps into Riemannian manifolds 162
Energy of maps into Riemannian polyhedra 173
The volume of a map 176
10. Holder continuity of energy minimizers 178
The case of a target of nonpositive curvature 179
Proof of Theorem 10.1 189
The case of a target of upper bounded curvature 192
11. Existence of energy minimizers 198
The case of free homotopy 200
The Dirichlet problem relative to a homotopy class 206
The ordinary Dirichlet problem 208
The case where the target is a Riemannian manifold 211
The case of 2 dimensional manifold domains 211
Questions and remarks 213
12. Harmonic maps. Totally geodesic maps 217
A concept of harmonic map 217
Weakly harmonic maps into a Riemannian manifold 221
Holder continuity revisited 230
Totally geodesic maps 233
Geodesies as harmonic maps 236
Jensen s inequality for maps 241
Harmonic maps from a 1 dimensional Riemannian polyhedron 243
13. Harmonic morphisms 247
Harmonic morphisms between harmonic spaces 247
Harmonic morphisms between Riemannian polyhedra 249
Harmonic morphisms into Riemannian manifolds 251
14. Appendix: Energy according to Korevaar Schoen 259
Subpartitioning Lemma 259
Directional energies 261
Trace maps 262
15. Appendix: Minimizers with small energy decay
(By T. Serbinowski) 264
Introduction and results 264
Embedding Y into an NPC cone 265
Holder continuity of the minimizer 268
viii Contents
Proof of Theorem 15.1 273
Lipschitz continuity of the minimizer 275
Bibliography 277
Special symbols 291
Index 294
|
| any_adam_object | 1 |
| author | Eells, James 1926-2007 Fuglede, Bent 1925- |
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| dewey-raw | 514.74 |
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| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV013873715 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:53:34Z |
| institution | BVB |
| isbn | 0521773113 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009490607 |
| oclc_num | 248559266 |
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| owner | DE-384 DE-83 DE-11 DE-19 DE-BY-UBM |
| owner_facet | DE-384 DE-83 DE-11 DE-19 DE-BY-UBM |
| physical | XII, 296 S. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge tracts in mathematics |
| series2 | Cambridge tracts in mathematics |
| spelling | Eells, James 1926-2007 Verfasser (DE-588)11533050X aut Harmonic maps between Riemannian polyhedra J. Eells ; B. Fuglede 1. publ. Cambridge Cambridge Univ. Press 2001 XII, 296 S. txt rdacontent n rdamedia nc rdacarrier Cambridge tracts in mathematics 142 Harmonic maps Riemannian manifolds Harmonische Abbildung (DE-588)4023452-6 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Harmonische Abbildung (DE-588)4023452-6 s Riemannscher Raum (DE-588)4128295-4 s DE-604 Fuglede, Bent 1925- Verfasser (DE-588)1078975876 aut Cambridge tracts in mathematics 142 (DE-604)BV000000001 142 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009490607&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Eells, James 1926-2007 Fuglede, Bent 1925- Harmonic maps between Riemannian polyhedra Cambridge tracts in mathematics Harmonic maps Riemannian manifolds Harmonische Abbildung (DE-588)4023452-6 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)4023452-6 (DE-588)4128295-4 |
| title | Harmonic maps between Riemannian polyhedra |
| title_auth | Harmonic maps between Riemannian polyhedra |
| title_exact_search | Harmonic maps between Riemannian polyhedra |
| title_full | Harmonic maps between Riemannian polyhedra J. Eells ; B. Fuglede |
| title_fullStr | Harmonic maps between Riemannian polyhedra J. Eells ; B. Fuglede |
| title_full_unstemmed | Harmonic maps between Riemannian polyhedra J. Eells ; B. Fuglede |
| title_short | Harmonic maps between Riemannian polyhedra |
| title_sort | harmonic maps between riemannian polyhedra |
| topic | Harmonic maps Riemannian manifolds Harmonische Abbildung (DE-588)4023452-6 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Harmonic maps Riemannian manifolds Harmonische Abbildung Riemannscher Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009490607&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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