  We develop a method for preserving pseudoholomorphic curves in contact
3-manifolds under surgery along transverse links. This makes use of a
geometrically natural boundary value problem for holomorphic curves in a
3-manifold with stable Hamiltonian structure, where the boundary conditions are
defined by 1-parameter families of totally real surfaces. The technique is
applied here to construct a finite energy foliation for every closed
overtwisted contact 3-manifold.
