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 $\mathbb{B}^n$.
 Publication:

arXiv eprints
 Pub Date:
 February 2017
 arXiv:
 arXiv:1702.08032
 Bibcode:
 2017arXiv170208032A
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Differential Geometry
 EPrint:
 To appear in Proc. Amer. Math. Soc