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arXiv · 2211.05397

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

Giovanni Domenico Di Salvo·Tyson Ritter·Erlend F. Wold
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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.

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Giovanni Domenico Di Salvo, Tyson Ritter, Erlend F. Wold. 2022-11-10. Families of Proper Holomorphic Embeddings and Carleman-type Theorems with parameters. https://arxiv.org/abs/2211.05397

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