Compactness for punctured holomorphic curves

Abstract

Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder recently proved a general compactness result for moduli spaces of punctured holomorphic curves arising in symplectic field theory. In this paper we present an alternative proof of this result. The main idea is to determine a priori the levels at which holomorphic curves split, thus reducing the proof to two separate cases: long cylinders of small area, and regions with compact image. The second case requires a generalization of Gromov compactness for holomorphic curves with free boundary.

Sutures and contact homology I

“…Let J be an almost complex structure on R M tailored to .M ;˛ /, as defined in Section 3.1. In this section we show that the SFT compactness theorem for holomorphic curves in the symplectization of a closed contact manifold, proved by Bourgeois et al [3] and Cieliebak and Mohnke [6], extends to the case of Jholomorphic curves in R M . At the end of this section, we extend the compactness theorem for embedded contact homology of the last author [24] to R M in the case dim.M / D 3.…”

Symplectic Floer homology of area-preserving surface diffeomorphisms

“…Cylindrical contact homology Symplectic field theory and contact homology invariants (see Eliashberg, Givental and Hofer [10]) exist for mapping tori due to the existence of a Hamiltonian structure (see Bourgeois et al [3], Cieliebak and Mohnke [5] and Fabert [11]). In this setting, the "cylindrical mapping torus contact homology" splits as a direct sum…”

Lagrangian spheres in Del Pezzo surfaces

“…• A smooth quadric surface Q ⊂ CP 3 or, equivalently, a product CP 1 × CP 1 (thinking of the CP 1 factors as rulings on the quadric surface), • A blow-up, D n , of CP 2 at n points in general position for n < 8.…”