Suppose that (X,ω) is a symplectic manifold and that there exists a Liouville vector field V defined in a neighbourhood of and transverse to M = ∂X. Then V induces a contact form α = ιVω[mid ]M on M which determines the germ of ω along M. One should think of the contact manifold (M,ξ = ker α) as controlling the behaviour of ω ‘at infinity’. If V points out of X along M then we call (X,ω) a convex filling of (M,ξ), and if V points into X along M then we call (X,ω) a concave filling of (M,ξ).
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