Deformations of Noncompact Complex Curves and Meromorphic Envelopes of Spheres

We study the envelopes of meromorphy of neighborhoods of symplectically immersed two-spheres in complex K"ahler surfaces using the Gromov’s theory of pseudoholomorphic curves. The construction of a complete family of holomorphic deformations of a non-compact complex curve in a complex manifold, parametrized by a finite codimension analytic subset of a Banach ball, is given. The existence of this family is used to prove a generalization of Levi’s continuity principle, which is applied to describe envelopes of meromorphy.