Families of Proper Holomorphic Embeddings and Carleman-type Theorems with parameters

We solve the problem of simultaneously embedding properly holomorphically into ${\mathbb{C}}^2$ a whole family of $n$-connected domains ${\mathrm{\Omega }}_r\subset {\mathbb{P}}^1$ such that none of the components of ${\mathbb{P}}^1\setminus {\mathrm{\Omega }}_r$ reduces to a point, by constructing a continuous mapping $\mathrm{\Xi }\text{\hspace{0.17em}⁣}:{\bigcup }_r\{r\}\times {\mathrm{\Omega }}_r\to {\mathbb{C}}^2$ such that $\mathrm{\Xi }(r,\cdot )\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_r\hookrightarrow {\mathbb{C}}^2$ 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.