Proper holomorphic embeddings of Riemann surfaces with arbitrary   topology into $\mathbb{C}^2$
  We prove that given an open Riemann surface $N,$ there exists an open domain $M\subset N$ homeomorphic to $N$ which properly holomorphically embeds in $\mathbb{C}^2.$ Furthermore, $M$ can be chosen with hyperbolic conformal type. In particular, any open orientable surface admits a complex structure properly holomorphically embedding into $\mathbb{C}^2.$ 
Antonio Alarcon and Francisco J. Lopez
math.CV math.DG