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

## TL;DR

This paper constructs complete, dense, injective holomorphic immersions of complex lines into a2^2 and extends the results to closed complex submanifolds, demonstrating the existence of dense embeddings with completeness properties.

## Contribution

It introduces new constructions of complete, dense holomorphic immersions and embeddings of complex submanifolds into complex Euclidean spaces, generalizing previous results.

## Key findings

- Existence of complete, dense holomorphic immersions of a2 into a2^2.
- Extension of results to any closed complex submanifold of a2^n.
- Construction of dense embeddings within Runge domains in a2^n.

## 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 $\mathbb{B}^n$.