J-holomorphic curves and symplectic topology:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Mathematical Society
2004
|
| Series: | American Mathematical Society: Colloquium publications
52 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 669 S. |
| ISBN: | 0821834851 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface ix
Chapter 1. Introduction 1
1.1. Symplectic manifolds 1
1.2. Moduli spaces: regularity and compactness 4
1.3. Evaluation maps and pseudocycles 7
1.4. The Gromov Witten invariants 10
Chapter 2. J holomorphic Curves 17
2.1. Almost complex structures 17
2.2. The nonlinear Cauchy Riemann equations 19
2.3. Unique continuation 21
2.4. Critical points 26
2.5. Somewhere injective curves 30
2.6. The adjunction inequality 34
Chapter 3. Moduli Spaces and Transversality 37
3.1. Moduli spaces of simple curves 37
3.2. Transversality 46
3.3. A regularity criterion 54
3.4. Curves with pointwise constraints 58
3.5. Implicit function theorem 65
Chapter 4. Compactness 71
4.1. Energy 72
4.2. The bubbling phenomenon 75
4.3. The mean value inequality 80
4.4. The isoperimetric inequality 85
4.5. Removal of singularities 89
4.6. Convergence modulo bubbling 92
4.7. Bubbles connect 98
Chapter 5. Stable Maps 107
5.1. Stable maps 107
5.2. Gromov convergence 114
5.3. Gromov compactness 117
5.4. Uniqueness of the limit 125
5.5. Gromov compactness for stable maps 129
5.6. The Gromov topology 137
Chapter 6. Moduli Spaces of Stable Maps 143
V
vi CONTENTS
6.1. Simple stable maps 145
6.2. Transversality for simple stable maps 148
6.3. Transversality for evaluation maps 153
6.4. Semipositivity 156
6.5. Pseudocycles 159
6.6. Gromov Witten pseudocycles 165
6.7. The pseudocycle of graphs 169
Chapter 7. Gromov Witten Invariants 187
7.1. Counting pseudoholomorphic spheres 189
7.2. Variations on the definition 195
7.3. Counting pseudoholomorphic graphs 203
7.4. Rational curves in projective spaces 208
7.5. Axioms for Gromov Witten invariants 222
Chapter 8. Hamiltonian Perturbations 239
8.1. Trivial bundles 240
8.2. Locally Hamiltonian fibrations 246
8.3. Pseudoholomorphic sections 252
8.4. Pseudoholomorphic spheres in the fiber 259
8.5. The pseudocycle of sections 261
8.6. Counting pseudoholomorphic sections 266
Chapter 9. Applications in Symplectic Topology 275
9.1. Periodic orbits of Hamiltonian systems 276
9.2. Obstructions to Lagrangian embeddings 290
9.3. The nonsqueezing theorem 301
9.4. Symplectic 4 manifolds 307
9.5. The group of symplectomorphisms 320
9.6. Hofer geometry 328
9.7. Distinguishing symplectic structures 334
Chapter 10. Gluing 343
10.1. The gluing theorem 344
10.2. Connected sums of J holomorphic curves 347
10.3. Weighted norms 349
10.4. Cutoff functions 353
10.5. Construction of the gluing map 356
10.6. The derivative of the gluing map 365
10.7. Surjectivity of the gluing map 373
10.8. Proof of the splitting axiom 379
Chapter 11. Quantum Cohomology 387
11.1. The small quantum cohomology ring 388
11.2. The Gromov Witten potential 403
11.3. Four examples 409
11.4. The Seidel representation 432
11.5. Frobenius manifolds 443
Chapter 12. Floer Homology 451
CONTENTS vii
12.1. Floer s cochain complex 452
12.2. Ring structure 463
12.3. Poincare duality 467
12.4. Spectral invariants 469
12.5. The Seidel representation 478
12.6. Donaldson s quantum category 483
12.7. The symplectic vortex equations 487
Appendix A. Fredholm Theory 493
A.I. Fredholm theory 493
A.2. Determinant line bundles 495
A.3. The implicit function theorem 500
A.4. Finite dimensional reduction 506
A.5. The Sard Smale theorem 508
Appendix B. Elliptic Regularity 511
B.I. Sobolev spaces 511
B.2. The Calderon Zygmund inequality 523
B.3. Regularity for the Laplace operator 530
B.4. Elliptic bootstrapping 533
Appendix C. The Riemann Roch Theorem 541
C.I. Cauchy Riemann operators 541
C.2. Elliptic estimates 546
C.3. The boundary Maslov index (by Joel Robbin) 552
C.4. Proof of the Riemann Roch theorem 558
C.5. The Riemann mapping theorem 562
C.6. Nonsmooth bundles 564
C.7. Almost complex structures 565
Appendix D. Stable Curves of Genus Zero 569
D.I. Mobius transformations and cross ratios 569
D.2. Trees, labels, and splittings 572
D.3. Stable curves 579
D.4. An embedding 582
D.5. The Grothendieck Knudsen compactification 587
D.6. The GK topology 592
D.7. Examples 595
Appendix E. Singularities and Intersections
(written with Laurent Lazzarini) 601
E.I. The main results 602
E.2. Positivity of intersections 606
E.3. Integrability 612
E.4. The Hartman Wintner theorem 616
E.5. Local behaviour 621
E.6. Contact between branches 626
E.7. Singularities of J holomorphic curves 634
Bibliography 643
viii CONTENTS
List of Symbols 655
Index 659
|
| any_adam_object | 1 |
| author | McDuff, Dusa |
| author_GND | (DE-588)1015895271 |
| author_facet | McDuff, Dusa |
| author_role | aut |
| author_sort | McDuff, Dusa |
| author_variant | d m dm |
| building | Verbundindex |
| bvnumber | BV019359539 |
| classification_rvk | SK 780 |
| ctrlnum | (OCoLC)440699453 (DE-599)BVBBV019359539 |
| dewey-full | 516.36 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.36 |
| dewey-search | 516.36 |
| dewey-sort | 3516.36 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T19:58:27Z |
| institution | BVB |
| isbn | 0821834851 |
| language | English |
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| physical | XII, 669 S. |
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| spelling | McDuff, Dusa Verfasser aut J-holomorphic curves and symplectic topology Dusa McDuff ; Dietmar Salamon Providence, R.I. American Mathematical Society 2004 XII, 669 S. txt rdacontent n rdamedia nc rdacarrier American Mathematical Society: Colloquium publications 52 Pseudoholomorphe Funktion (DE-588)4176138-8 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Symplektische Mannigfaltigkeit (DE-588)4290704-4 gnd rswk-swf Holomorphe Kurve (DE-588)4160476-3 gnd rswk-swf Holomorphie (DE-588)4160484-2 gnd rswk-swf Pseudoholomorphe Funktion (DE-588)4176138-8 s Symplektische Mannigfaltigkeit (DE-588)4290704-4 s DE-604 Holomorphie (DE-588)4160484-2 s Differentialgeometrie (DE-588)4012248-7 s Holomorphe Kurve (DE-588)4160476-3 s 1\p DE-604 Salamon, Dietmar 1953- Sonstige (DE-588)1015895271 oth American Mathematical Society: Colloquium publications 52 (DE-604)BV035417609 52 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012823304&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | McDuff, Dusa J-holomorphic curves and symplectic topology American Mathematical Society: Colloquium publications Pseudoholomorphe Funktion (DE-588)4176138-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd Symplektische Mannigfaltigkeit (DE-588)4290704-4 gnd Holomorphe Kurve (DE-588)4160476-3 gnd Holomorphie (DE-588)4160484-2 gnd |
| subject_GND | (DE-588)4176138-8 (DE-588)4012248-7 (DE-588)4290704-4 (DE-588)4160476-3 (DE-588)4160484-2 |
| title | J-holomorphic curves and symplectic topology |
| title_auth | J-holomorphic curves and symplectic topology |
| title_exact_search | J-holomorphic curves and symplectic topology |
| title_full | J-holomorphic curves and symplectic topology Dusa McDuff ; Dietmar Salamon |
| title_fullStr | J-holomorphic curves and symplectic topology Dusa McDuff ; Dietmar Salamon |
| title_full_unstemmed | J-holomorphic curves and symplectic topology Dusa McDuff ; Dietmar Salamon |
| title_short | J-holomorphic curves and symplectic topology |
| title_sort | j holomorphic curves and symplectic topology |
| topic | Pseudoholomorphe Funktion (DE-588)4176138-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd Symplektische Mannigfaltigkeit (DE-588)4290704-4 gnd Holomorphe Kurve (DE-588)4160476-3 gnd Holomorphie (DE-588)4160484-2 gnd |
| topic_facet | Pseudoholomorphe Funktion Differentialgeometrie Symplektische Mannigfaltigkeit Holomorphe Kurve Holomorphie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012823304&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035417609 |
| work_keys_str_mv | AT mcduffdusa jholomorphiccurvesandsymplectictopology AT salamondietmar jholomorphiccurvesandsymplectictopology |