A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (ℂ*)2

Let X be an open Riemann surface. We prove an Oka property on the approximation and interpolation of continuous maps X \to (\mathbb{C}^{*})^{2} by proper holomorphic embeddings, provided that we permit a smooth deformation of the complex structure on X outside a certain set. This generalises and strengthens a recent result of Alarcón and López. We also give a Forstnerič–Wold theorem for proper holomorphic embeddings (with respect to the given complex structure) of certain open Riemann surfaces into {(\mathbb{C}^{*})^{2}} .