Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets

Abstract

We 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.

Proper holomorphic embeddings with small limit sets

“…(There are algebraic open Riemann surfaces which do not embed as smooth proper affine curves in C 2 .) Corollary 1.2 gives proper holomorphic embeddings f : X → C 3 whose images are arbitrarily close to the subspace C 2 × {0} in the Fubini-Study metric on CP 3 , and f (X) ∞ = CP 1 .It was recently shown by Drinovec Drnovšek and Forstnerič[8, Theorem 1.3]…”