A few remarks about symplectic filling

We show that any compact symplectic manifold (W, omega) with boundary embeds as a domain into a closed symplectic manifold, provided that there exists a contact plane xi on partial derivativeW which is weakly compatible with omega, i.e. the restriction omega\xi does not vanish and the contact orientation of partial derivativeW and its orientation as the boundary of the symplectic manifold W coincide. This result provides a useful tool for new applications by Ozsvath-Szabo of Seiberg-Witten Floer homology theories in three-dimensional topology and has helped complete the Kronheimer-Mrowka proof of Property P for knots.