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

Abstract

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