Preface to the second edition
Preface
Chapter 1. Introduction
1.1. Symplectic manifolds
1.2. Moduli spaces: regularity and compactness
1.3. Evaluation maps and pseudocycles
1.4. The Gromov-Witten invariants
1.5. Applications and further developments
Chapter 2. J-holomorphic Curves
2.1. Almost complex structures
2.2. The nonlinear CauchyoRiemann equations
2.3. Unique continuation
2.4. Critical points
2.5. Somewhere injective curves
2.6. The adjunction inequality
Chapter 3. Moduli Spaces and Transversality
3.1. Moduli spaces of simple curves
3.2. Transversality
3.3. A regularity criterion
3.4. Curves with pointwise constraints
3.5. Implicit function theorem
Chapter 4. Compactness
4.1. Energy
4.2. The bubbling phenomenon
4.3. The mean value inequality
4.4. The isoperimetric inequality
4.5. Removal of singularities
4.6. Convergence modulo bubbling
4.7. Bubbles connect
Chapter 5. Stable Maps
5.1. Stable maps
5.2. Gromov convergence
5.3. Gromov compactness
5.4. Uniqueness of the limit
5.5. Gromov compactness for stable maps
5.6. The Gromov topology
Chapter 6. Moduli Spaces of Stable Maps
6.1. Simple stable maps
6.2. Transversality for simple stable maps
6.3. Transversality for evaluation maps
6.4. Semipositivity
6.5. Pseudocycles
6.6. Gromov-Witten pseudocycles
6.7. The pseudocycle of graphs
Chapter 7. Gromov-Witten Invariants
7.1. Counting pseudoholomorphic spheres
7.2. Variations on the definition
7.3. Counting pseudoholomorphic graphs
7.4. Rational curves in projective spaces
7.5. Axioms for Gromov-Witten invariants
Chapter 8. Hamiltonian Perturbations
8.1. Trivial bundles
8.2. Locally Hamiltonian fibrations
8.3. Pseudoholomorphic sections
8.4. Pseudoholomorphic spheres in the fiber
8.5. The pseudocycle of sections
8.6. Counting pseudoholomorphic sections
Chapter 9. Applications in Symplectic Topology
9.1. Periodic orbits of Hamiltonian systems
9.2. Obstructions to Lagrangian embeddings
9.3. The nonsqueezing theorem
9.4. Symplectic 4-manifolds
9.5. The group of symplectomorphisms
9.6. Hofer geometry
9.7. Distinguishing symplectic structures
Chapter 10. Gluing
10.1. The gluing theorem
10.2. Connected sums of J-holomorphic curves
10.3. Weighted norms
10.4. Cutoff functions
10.5. Construction of the gluing map
10.6. The derivative of the gluing map
10.7. Surjectivity of the gluing map
10.8. Proof of the splitting axiom
10.9. The gluing theorem revisited
Chapter 11. Quantum Cohomology
11.1. The small quantum cohomology ring
11.2. The Gromov-Witten potential
11.3. Four examples
11.4. The Seidel representation
11.5. Frobenius manifolds
Chapter 12. Floer Homology
12.1. Floer's cochain complex
12.2. Ring structure
12.3. Poincare duality
12.4. Spectral invariants
12.5. The Seidel representation
12.6. Donaldson's quantum category
12.7. The symplectic vortex equations
Appendix A. Fredholm Theory
A.1. Fredholm theory
A.2. Determinant line bundles
A.3. The implicit function theorem
A.4. Finite dimensional reduction
A.5. The Sard-Smale theorem
Appendix B. Elliptic Regularity
B.1. Sobolev spaces
B.2. The Calderon-Zygmund inequality
B.3. Regularity for the Laplace operator
B.4. Elliptic bootstrapping
Appendix C. The Riemann-Roch Theorem
C.1. Cauchy-Riemann operators
C.2. Elliptic estimates
C.3. The boundary Maslov index (by Joel Robbin)
C.4. Proof of the Riemann-Roch theorem
C.5. The Riemann mapping theorem
C.6. Nonsmooth bundles
C.7. Almost complex structures
Appendix D. Stable Curves of Genus Zero
D.1. MSbius transformations and cross ratios
D.2. Trees, labels, and splittings
D.3. Stable curves
D.4. The Grothendieck-Knudsen manifold
D.5. The Gromov topology
D.6. Cohomology
D.7. Examples
Appendix E. Singularities and Intersections(written with Laurent Lazzarini)
E.1. The main results
E.2. Positivity of intersections
E.3. Integrability
E.4. The Hartman-Wintner theorem
E.5. Local behaviour
E.6. Contact between branches
E.7. Singularities of J-holomorphic curves
Bibliography
List of Symbols
Index
