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

TL;DR

This paper develops a parametric approach to properly embed a family of complex domains into b5^2, extending classical approximation theorems to a parameter-dependent setting with applications to holomorphic embeddings.

Contribution

It introduces a parametric version of the Anderse9n-Lempert technique and Carleman's Theorem, enabling simultaneous embeddings of domain families into b5^2.

Findings

Constructed a continuous family of proper holomorphic embeddings.

Extended classical approximation theorems to parameter-dependent contexts.

Provided a method for embedding families of domains into complex Euclidean spaces.

Abstract

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