Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets

Abstract

AbstractWe show that if E is a closed convex set in $${\mathbb {C}}^n$$ C n $$(n>1)$$ ( n > 1 ) contained in a closed halfspace H such that $$E\cap bH$$ E ∩ b H is nonempty and bounded, then the concave domain $$\Omega = {\mathbb {C}}^n{\setminus } E$$ Ω = C n \ E contains images of proper holomorphic maps $$f:X\rightarrow {\mathbb {C}}^n$$ f : X → C n from any Stein manifold X of dimension $$<n$$ < n , with approximation of a given map on closed compact subsets of X. If in addition $$2\dim X+1\le n$$ 2 dim X + 1 ≤ n then f can be chosen an embedding, and if $$2\dim X=n$$ 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.