Weak analytic hyperbolicity of complements of generic surfaces of high   degree in projective 3-space
  In this article we prove that every entire curve in the complement of a generic hypersurface of degree $d\geq 586$ in $\mathbb{P}_{\mathbb{C}}^{3}$ is algebraically degenerate i.e there exists a proper subvariety which contains the entire curve. 
Erwan Rousseau