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

TL;DR

This paper constructs complete complex hypersurfaces within the unit ball of complex Euclidean space, with control over their topology, by developing new embedding techniques for complex hypersurfaces and curves.

Contribution

It introduces a method to embed complete complex hypersurfaces into the unit ball with prescribed topology, a novel achievement in complex geometry.

Findings

Existence of complete proper holomorphic embeddings of certain domains into the unit ball.

Construction of complete bounded embedded complex hypersurfaces with controlled topology.

First examples of such hypersurfaces with arbitrary finite topology.

Abstract

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.