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We construct holomorphically varying families of Fatou-Bieberbach domainswith 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 recentresult of Yuta Kusakabe to the effect that the complement $\mathbb C^n\setminusK$ 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 Steinmanifold with the density property.

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