AN EMBEDDING OF C IN C^{2}

WITH HYPERBOLIC COMPLEMENT

Gregery T. Buzzard John Erik Fornaess

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

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