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 $\Omega$ 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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- June 1995
- DOI:
- 10.48550/arXiv.math/9506211
- arXiv:
- arXiv:math/9506211
- Bibcode:
- 1995math......6211B
- Keywords:
-
- Mathematics - Dynamical Systems;
- Mathematics - Complex Variables