Strongly fillable contact manifolds and J-holomorphic foliations

Abstract

We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of $T^3$ similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of $T^3$ are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured $J$-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on $T^{*}{T}^2$ is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result [G] for contact manifolds with positive Giroux torsion