Embedding holomorphic discs through discrete sets

Abstract

The most interesting case is n = 2. For n > 3 this was proved by a different method in [16] for f~ = n and in [9] for all convex domains O n. In the case when f2 = C (n > 3) one can in addition prescribe a discrete set {~/} and require that f ( ~ j ) = z j for all j [16]. For n = 2 the methods of [9] and [16] only give proper holomorphic immersions of the disc through a given discrete set in 12. The main problem of course is that, in dimension two, one cannot remove self-intersections o f complex curves by small deformations. We do not know whether our Theorem holds for non-Runge pseudoconvex domains in cn; our methods do not seem to extend to this case. In this direction it was proved in [6] that for every finite subset Z in an arbitrary connected pseudoconvex domain f2 n (n > 1) there exist proper holomorphic mappings f : D ~ D o f the disc into 12 such that Z c f ( D ) . It is likely that a refinement of the construction in [6] gives proper holomorphic immersions f : D ~ f2 whose image contains a given discrete subset of 12. An example