Proper Holomorphic Embeddings of Riemann Surfaces with Arbitrary Topology into ℂ2

Abstract

We prove that given an open Riemann surface $\mathcal{N}$ of arbitrary (finite or infinite) topology, there exists an open domain $\mathcal{M}\subset \mathcal{N}$ homeomorphic to $\mathcal{N}$ which properly holomorphically embeds in ℂ2. Furthermore, $\mathcal{M}$ can be chosen with hyperbolic conformal type. In particular, any open orientable surface M admits a complex structure $\mathcal{C}$ such that $(M,\mathcal{C})$ can be properly holomorphically embedded into ℂ2.