Explicit Concave Fillings of Contact Three-manifolds

When (M, ξ) is a contact 3-manifold we say that a compact symplectic 4-manifold (X,ω) is a concave filling of (M, ξ) ifM = −∂X and if there exists a Liouville vector field V defined on a neighborhood of M , transverse to M and pointing in to X , such that ξ is the kernel of ıV ω restricted toM . We give explicit, handleby-handle constructions of concave fillings of all closed, oriented, contact 3-manifolds.