# Proper holomorphic immersions into Stein manifolds with the density property

## TL;DR

This paper proves that any Stein manifold can be properly holomorphically immersed into Stein manifolds with the density property, generalizing classical results and extending the scope of holomorphic immersion theory.

## Contribution

It establishes the existence of proper holomorphic immersions from Stein manifolds into Stein manifolds with the density property for all dimensions, generalizing previous special cases.

## Key findings

- Every Stein manifold admits a proper holomorphic immersion into Stein manifolds with the density property.
- The result generalizes classical theorems for immersions into complex Euclidean spaces.
- Complements previous embedding theorems by broadening the target class of Stein manifolds.

## Abstract

In this paper we prove that every Stein manifold $S$ admits a proper holomorphic immersion into any Stein manifold $X$ of dimension $2\mathrm{dim}S$ enjoying the density property or the volume density property. The case $\mathrm{dim}S=1$ was proved beforehand by Andrist and Wold (Ann. Inst. Fourier (Grenoble), 64(2):681-697, 2014). This result generalizes the classical theorem of Bishop and Narasimhan for immersions to $\mathbb{C}^n$ with $n=2\mathrm{dim}S$, and it complements the proper embedding theorem proved by Andrist et al. when $\mathrm{dim}X>2\mathrm{dim}S$ (J. Anal. Math., 130:135-150, 2016).