  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 the 3-torus 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 the 3-torus 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 the cotangent bundle of the 2-torus is contractible,
and to define an obstruction to strong fillability that yields a
non-gauge-theoretic proof of Gay's recent nonfillability result for contact
manifolds with positive Giroux torsion.
