An embedding of ${\Bbb C}$ in ${\Bbb C}^2$ with hyperbolic complement
Gregery T. Buzzard and John Erik Fornaess
Let $X$ be a closed, $1$-dimensional, complex subvariety of ${\Bbb C}^2$ and let $\overline{\Bbb B}$ be a closed ball in ${\Bbb C}^2 - X$. Then there exists a Fatou-Bieberbach domain $\Omega$ with $X \subseteq \Omega \subseteq {\Bbb C}^2 - \overline{\Bbb B}$ and a biholomorphic map $\Phi: \Omega \rightarrow {\Bbb C}^2$ such that ${\Bbb C}^2 - \Phi(X)$ is Kobayashi hyperbolic. As corollaries, there is an embedding of the plane in ${\Bbb C}^2$ whose complement is hyperbolic, and there is a nontrivial Fatou-Bieberbach domain containing any finite collection of complex lines.