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

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

1995-06-20

arxiv.org/abs/math/9506211v1

Mathematics

Dynamical Systems

Scientific paper

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 $\Omega$ with $X \subseteq \Omega \subseteq \CC^2 - \ol{\BB}$ and a biholomorphic map $\Phi: \Omega \ra \CC^2$ such that $\CC^2 - \Phi(X)$ 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.

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