AbstractGiven a
closed complex hypersurface {Z\subset\mathbb{C}^{N+1}} ({N\in\mathbb{N}}) and a compact subset {K\subset Z}, we prove the existence of a pseudoconvex Runge domain D in Z such that {K\subset D} and there is a complete proper holomorphic embedding from D into the unit ball of {\mathbb{C}^{N+1}}.
For {N=1}, we derive the existence of complete properly embedded complex curves in the unit ball of {\mathbb{C}^{2}}, with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc {\mathbb{D}\subset\mathbb{C}} into the unit ball of {\mathbb{C}^{2}}.
These are the first known examples of complete bounded embedded complex hypersurfaces in {\mathbb{C}^{N+1}} with any control on the topology.
Implications of Stahl's theorems to holomorphic embedding Pt. I: Theoretical convergence
