Interpolation by holomorphic automorphisms and embeddings in Cn

Abstract

Let n > 1 and let C n denote the complex n-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings F:C n →C n and for holomorphic automorphisms of C n on discrete subsets of C n.We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds into C n.For each closed complex submanifold (or subvariety) M ⊂ C n of complex dimension m < n we construct a domain Ω ⊂C n containing M and a biholomorphic map F: Ω → C n onto C n with J F ≡ 1such that F(M) intersects the image of any nondegenerate entire map G:C n−m →C n at infinitely many points. If m = n − 1, we construct F as above such that C n ∖F(M) is hyperbolic. In particular, for each m ≥ 1we construct proper holomorphic embeddings F:C m →C m−1 such that the complement C m+1 ∖F(C m )is hyperbolic.