Stability of hyperbolic imbeddedness and construction of examples

Methods are worked out for constructing smooth hyperbolic curves $\Gamma\subset\mathbf{CP}^2$ and surfaces $H\subset\mathbf{CP}^3$ with hyperbolically imbedded complements, and the methods are then used to construct examples of such curves with least possible degree 5. The existence of these curves agrees well with the 1970 conjecture of Kobayashi. It is proved that the sets of such curves and surfaces are open (in the classical topology). The proofs are based on tests obtained for stability of hyperbolicity and of hyperbolic imbeddedness of analytic subsets of complex manifolds under perturbations that can in general reconstruct the topology.
Bibliography: 18 titles.