An embedding of {\bf C} in {\bf C}$^2$ with hyperbolic complement

Abstract

Let $X$ be a closed, $1$-dimensional, complex subvariety of $\CC^2$ and let $\ol{\BB}$ be a closed ball in $\CC^2 - X$. Then there exists a Fatou-Bieberbach domain $Ω$ with $X \subseteq Ω\subseteq \CC^2 - \ol{\BB}$ and a biholomorphic map $Φ: Ω\ra \CC^2$ such that $\CC^2 - Φ(X)$ is Kobayashi hyperbolic. As corollaries, there is an embedding of the plane in $\CC^2$ whose complement is hyperbolic, and there is a nontrivial Fatou-Bieberbach domain containing any finite collection of complex lines.