Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets

TL;DR

This paper demonstrates that certain convex sets in complex Euclidean spaces allow for proper holomorphic maps from Stein manifolds to avoid these sets, with approximation, embedding, and interpolation properties depending on dimension constraints.

Contribution

It establishes conditions under which proper holomorphic maps from Stein manifolds can be constructed to avoid specific convex sets in complex Euclidean spaces, including approximation, embedding, and interpolation results.

Findings

Proper holomorphic maps exist avoiding convex sets in complex spaces.

Maps can be approximated on compact subsets of Stein manifolds.

Embedding and immersion properties depend on dimension relations.

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 $\Omega = \mathbb C^n\setminus E$ 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.