Complete embedded complex curves in the ball of $\mathbb{C}^2$ can have   any topology
  In this paper we prove that the unit ball $\mathbb{B}$ of $\mathbb{C}^2$ admits complete properly embedded complex curves of any given topological type. Moreover, we provide examples containing any given closed discrete subset of $\mathbb{B}$. 
Antonio Alarcon and Josip Globevnik
math.CV math.DG