Oka properties of ball complements

Abstract

Let $n>1$ be an integer. We prove that holomorphic maps from Stein manifolds $X$ of dimension $<n$ to the complement ${\mathbb{C}}^n\setminus L$ of a compact convex set $L\subset {\mathbb{C}}^n$ satisfy the basic Oka property with approximation and interpolation. If $L$ is polynomially convex then the same holds when $2\dim X<n$. We also construct proper holomorphic maps, immersions and embeddings $X\to {\mathbb{C}}^n$ with additional control of the range, thereby extending classical results of Remmert, Bishop and Narasimhan.

Version 6 has been withdrawn because it is no longer relevant. The proofs of Theorem 1, Corollary 2 and Theorem 15 in the published version (v5) are incomplete in the special case when $L$ is polynomially convex (and not holomorphically contractible) and ${\dim }_{R}X=n$. However, these proofs can be completed by applying Lemma 3.1 in the new preprint arXiv:1703.08594

Cited by in corpus (11)

• Proper holomorphic embeddings into Stein manifolds with the density property
• Recent developments on Oka manifolds
• The first thirty years of Andersen-Lempert theory
• Proper holomorphic immersions into Stein manifolds with the density property
• Euclidean domains in complex manifolds
• Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
• Flexible domains for minimal surfaces in Euclidean spaces
• Fatou-Bieberbach domains in ${\mathbb{C}}^n\setminus {\mathbb{R}}^k$
• Proper holomorphic curves attached to domains
• Proper maps of ball complements & differences and rational sphere maps
• Proper holomorphic embeddings with small limit sets