  We apply the method of filling with holomorphic discs to a 4-dimensional
symplectic cobordism with the standard contact 3-sphere as a convex boundary
component. We establish the following dichotomy: either the cobordism is
diffeomorphic to a ball, or there is a periodic Reeb orbit of quantifiably
short period in the concave boundary of the cobordism. This allows us to give a
unified treatment of various results concerning Reeb dynamics on contact
3-manifolds, symplectic fillability, the topology of symplectic cobordisms,
symplectic non-squeezing, and the non-existence of exact Lagrangian surfaces in
standard symplectic 4-space.
