Compactness for embedded pseudoholomorphic curves in 3-manifolds

Abstract

We prove a compactness theorem for holomorphic curves in 4-dimensional symplectizations that have embedded projections to the underlying 3-manifold. It strengthens the cylindrical case of the SFT compactness theorem [BEH+ 03] by using intersection theory to show that degenerations of such sequences never give rise to multiple covers or nodes, so transversality is easily achieved. This has application to the theory of stable finite energy foliations introduced in [HWZ03], and also suggests a new approach to defining SFT-type invariants for contact 3manifolds, or more generally, 3-manifolds with stable Hamiltonian structures.