Filling by holomorphic curves in symplectic 4-manifolds

We develop a general framework for embedded (immersed) J-holomorphic curves and a systematic treatment of the theory of filling by holomorphic curves in 4-dimensional symplectic manifolds. In particular, a deformation theory and an intersection theory for J-holomorphic curves with boundary are developed. Bishop's local filling theorem is extended to almost complex manifolds. Existence and uniqueness of global fillings are given complete proofs. Then they are extended to the situation with nontrivial J-holomorphic spheres, culminating in the construction of singular fillings.