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

Abstract

AbstractWe solve the problem of simultaneously embedding properly holomorphically into $${\mathbb {C}}^2$$ C 2 a whole family of n-connected domains $$\Omega _r\subset \mathbb P^1$$ Ω r ⊂ P 1 such that none of the components of $$\mathbb P^1\setminus \Omega _r$$ P 1 \ Ω r reduces to a point, by constructing a continuous mapping $$\Xi :\bigcup _r\{r\}\times \Omega _r\rightarrow {\mathbb {C}}^2$$ Ξ : ⋃ r { r } × Ω r → C 2 such that $$\Xi (r,\cdot ):\Omega _r\hookrightarrow {\mathbb {C}}^2$$ Ξ ( r , · ) : Ω r ↪ 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.