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

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$inplaceof$$\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 $\mathrm{\Omega }\subset X$ containing $K$ which admits a complete holomorphic embedding $\mathrm{\Omega }\to {\mathbb{B}}^n$ whose image is dense in ${\mathbb{B}}^n$.