Complete densely embedded complex lines in $\mathbb {C}^2$

Abstract:

In this paper we construct a complete injective holomorphic immersion $\mathbb {C}\to \mathbb {C}^2$ whose image is dense in $\mathbb {C}^2$. The analogous result is obtained for any closed complex submanifold $X\subset \mathbb {C}^n$ for $n>1$ in place of $\mathbb {C}\subset \mathbb {C}^2$. We also show that if $X$ intersects the unit ball $\mathbb {B}^n$ of $\mathbb {C}^n$ and $K$ is a connected compact subset of $X\cap \mathbb {B}^n$, then there is a Runge domain $\Omega \subset X$ containing $K$ which admits a complete injective holomorphic immersion $\Omega \to \mathbb {B}^n$ whose image is dense in $\mathbb {B}^n$.