A construction of complete complex hypersurfaces in the ball with control on the topology

Given 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.