J-holomorphic curves and symplectic topology: The essential reference for symplectic geometers in dj

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 T3 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 T3 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*T2 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

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