arXiv subjects
Publications and source records attributed to Giovanni Domenico Di Salvo.
We present a construction of a proper holomorphic embedding $f\colon \Bbb P^1\setminus C\hookrightarrow \Bbb C^2$, where C is a Cantor set obtained by removing smaller and smaller vertical and horizontal strips from a square of side 2, allowing to realize it to have Lebesgue measure arbitrarily close to 4.
We clarify the details of a cryptical paper by Orevkov in which a construction of a proper holomorphic embedding $γ\colon\Bbb P^1\setminus{C}\hookrightarrow\Bbb C^2$ is performed; in particular, it is proved that such a construction can be done to get the Cantor set $C$ to have zero Hausdorff dimension.
We display four approximation theorems for manifold-valued mappings. The first one approximates holomorphic embeddings on pseudoconvex domains in $\Bbb C^n$ with holomorphic embeddings with dense images. The second theorem approximates holomorphic mappings on complex manifolds with bounded images with holomorphic mappings with dense images. The last two theorems work the other way around, constructing (in different settings) sequences of holomorphic mappings (embeddings in the first one) converging to a mapping with dense image defined on a given compact minus certain points (thus in general not holomorphic).
We solve the problem of simultaneously embedding properly holomorphically into $\Bbb C^2$ a whole family of $n$-connected domains $Ω_r\subset\Bbb P^1$ such that none of the components of $\Bbb P^1\setminusΩ_r$ reduces to a point, by constructing a continuous mapping $Ξ\colon\bigcup_r\{r\}\timesΩ_r\to\Bbb C^2$ such that $Ξ(r,\cdot)\colonΩ_r\hookrightarrow\Bbb C^2$ 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.