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 \(2\dim X \le n\). We also construct proper holomorphic maps, immersions and embeddings \(X\rightarrow \mathbb {C}^n\) with additional control of the range, thereby extending classical results of Remmert, Bishop and Narasimhan.