Runge tubes in Stein manifolds with the density property

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.

In this version we added Remark 1.6 and Problem 1.7. The paper will appear in Proc. Amer. Math. Soc

References in corpus (2)

• Criteria for the density property of complex manifolds
• Proper holomorphic embeddings into Stein manifolds with the density property

Cited by in corpus (3)

• Recent developments on Oka manifolds
• The first thirty years of Andersen-Lempert theory
• Automorphisms of ${\mathbb{C}}^k$ with an invariant non-recurrent attracting Fatou component biholomorphic to $\mathbb{C}\times ({\mathbb{C}}^{\ast }{)}^{k-1}$