Holomorphic families of Fatou-Bieberbach domains and applications to Oka manifolds

We construct holomorphically varying families of Fatou-Bieberbach domains with given centres in the complement of any compact polynomially convex subset $K$ of $mathbb C^n$ for $n>1$. This provides a simple proof of the recent result of Yuta Kusakabe to the effect that the complement $mathbb C^nsetminus K$ of any polynomially convex subset $K$ of $mathbb C^n$ is an Oka manifold. The analogous result is obtained with $mathbb C^n$ replaced by any Stein manifold with the density property.