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arXiv · 2301.01268

Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets

Barbara Drinovec Drnovsek·Franc Forstneric
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Abstract

We show that if $E$ is a closed convex set in $\mathbb C^n$ $(n>1)$ contained in a closed halfspace $H$ such that $E\cap bH$ is nonempty and bounded, then the concave domain $Ω= \mathbb C^n\setminus E$ contains images of proper holomorphic maps $f:X\to \mathbb C^n$ from any Stein manifold $X$ of dimension $<n$, with approximation of a given map on closed compact subsets of $X$. If in addition $2\dim X+1\le n$ then $f$ can be chosen an embedding, and if $2\dim X=n$ then it can be chosen an immersion. Under a stronger condition on $E$ we also obtain the interpolation property for such maps on closed complex subvarieties.

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Barbara Drinovec Drnovsek, Franc Forstneric. 2023-01-03. Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets. https://doi.org/10.1007/s12220-023-01222-z

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