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.
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Chris Wendl, Compactness for embedded pseudoholomorphic curves in 3-manifolds. J. Eur. Math. Soc. 12 (2010), no. 2, pp. 313–342DOI 10.4171/JEMS/199