Families of Proper Holomorphic Embeddings and Carleman-Type Theorem with parameters

We solve the problem of simultaneously embedding properly holomorphically into $${\mathbb {C}}^2$$ a whole family of n-connected domains $$\Omega _r\subset \mathbb P^1$$ such that none of the components of $$\mathbb P^1\setminus \Omega _r$$ reduces to a point, by constructing a continuous mapping $$\Xi :\bigcup _r\{r\}\times \Omega _r\rightarrow {\mathbb {C}}^2$$ such that $$\Xi (r,\cdot ):\Omega _r\hookrightarrow {\mathbb {C}}^2$$ is a proper holomorphic embedding for every r. To this aim, a parametric version of both the Andersén–Lempert procedure and Carleman’s Theorem is formulated and proved.