  We study holomorphic spheres in certain symplectic cobordisms and derive
information about periodic Reeb orbits in the concave end of these cobordisms
from the non-compactness of the relevant moduli spaces. We use this to confirm
the strong Weinstein conjecture (predicting the existence of null-homologous
Reeb links) for various higher-dimensional contact manifolds, including contact
type hypersurfaces in subcritical Stein manifolds and in some cotangent
bundles. The quantitative character of this result leads to the definition of a
symplectic capacity.
