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Let $X$ be an open Riemann surface. We prove an Oka property on theapproximation and interpolation of continuous maps $X \to (\mathbb{C}^*)^2$ byproper holomorphic embeddings, provided that we permit a smooth deformation ofthe complex structure on $X$ outside a certain set. This generalises andstrengthens a recent result of Alarcon and Lopez. We also give aForstneric-Wold theorem for proper holomorphic embeddings (with respect to thegiven complex structure) of certain open Riemann surfaces into$(\mathbb{C}^*)^2$.

A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into $(\mathbb{C}^*)^2$