Runge tubes in Stein manifolds with the density property

TL;DR

This paper provides a simple proof for the existence of Runge tubes in Stein manifolds with the density property, demonstrating their abundance and embedding properties for algebraic submanifolds.

Contribution

It introduces a straightforward proof of Runge tubes' existence in Stein manifolds with the density property, extending classical results in complex analysis.

Findings

Runge tubes exist abundantly in Stein manifolds with the density property.

Normal bundles of certain algebraic submanifolds can be holomorphically embedded as Runge domains.

The embedding agrees with the inclusion on the zero section of the normal bundle.

Abstract

In this paper we give a very simple proof of the existence and plenitude of Runge tubes in $\mathbb C^n$ $(n>1)$ and, more generally, in Stein manifolds with the density property. We show in particular that for any algebraic submanifold $A$ of codimension at least two in a complex Euclidean space $\mathbb C^n$, the normal bundle of $A$ in $\mathbb C^n$ admits a holomorphic embedding onto a Runge domain in $\mathbb C^n$ which agrees with the inclusion map $A\hookrightarrow \mathbb C^n$ on the zero section.