Compactness for Embedded Pseudoholomorphic Curves in 3-manifolds

Details Compactness for Embedded Pseudoholomorphic Curves in 3-manifolds Compactness for Embedded Pseudoholomorphic Curves in 3-manifolds

2007-03-17

arxiv.org/abs/math/0703509v3

Mathematics

Symplectic Geometry

32 pages, 4 figures; v.3 includes some minor improvements to the exposition suggested by the referee

Scientific paper

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 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, and also suggests a new approach to defining SFT-type invariants for contact 3-manifolds, or more generally, 3-manifolds with stable Hamiltonian structures.

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