Non-Runge Fatou-Bieberbach Domains in Stein Manifolds with the Density Property

TL;DR

This paper develops methods to construct non-Runge Fatou-Bieberbach domains in Stein manifolds with the density property, expanding understanding of complex structures biholomorphic to Euclidean space or the manifold itself.

Contribution

It introduces new construction techniques for non-Runge Fatou-Bieberbach domains in Stein manifolds with the density property, including explicit examples.

Findings

Constructed two types of non-Runge Fatou-Bieberbach domains

Provided examples where the methods are applicable

Expanded the class of known biholomorphic domains in Stein manifolds

Abstract

Let $X$ be a Stein manifold with the density property. We present methods of constructing two kinds of non-Runge Fatou-Bieberbach domains in $X$, which by definition are proper open subsets of $X$ biholomorphic either to $\mathbb{C}^n$ or to $X$. For both kinds, we provide examples where our methods apply.