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 holomorphic embedding $\Omega \to {\mathbb{B}}^n$ whose image is dense in ${\mathbb{B}}^n$.

To appear in Proc. Amer. Math. Soc

References in corpus (1)

• Complete embedded complex curves in the ball of ${\mathbb{C}}^2$ can have any topology

Cited by in corpus (3)

• The first thirty years of Andersen-Lempert theory
• Embedded complex curves in the affine plane
• Dimer Models and Conformal Structures