On J-holomorphic Curves in Almost Complex Manifolds with Asymptotically Cylindrical Ends

Book excerpt: The compactification of moduli spaces of J-holomorphic curves in almost complex manifolds with cylindrical ends is crucial in Symplectic Field Theory. One natural generalization is to replace ``cylindrical'' by ``asymptotically cylindrical''. In this article we generalize the compactness results by Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder to this setting. As one application, we prove that the number of times that any smooth J-holomorphic curve passes through a fixed point in a closed symplectic manifold is bounded by a constant. The constant depends on the symplectic area, and does not depend on the domain Riemann surface and the map itself. Here J is any compatible smooth almost complex structure. In particular, we do not require J to be integrable. As another application, we study the relation between the moduli spaces of J-holomorphic polygons before and after the Lagrangian surgery established by Fukaya, Oh, Ohta and Ono in a more general setting and from a different viewpoint.