Fatou–Bieberbach domains in $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$

Abstract

We construct Fatou–Bieberbach domains in $\mathbb{C}^{n}$ for n>1 which contain a given compact set K and at the same time avoid a totally real affine subspace L of dimension < n, provided that K∪L is polynomially convex. By using this result, we show that the domain $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ for 1≤k< n enjoys the basic Oka property with approximation for maps from any Stein manifold of dimension < n.