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Interpolation by holomorphic automorphisms and embeddings in $${\mathbb{C}}^n$$. (English) Zbl 0963.32006
Summary: Let $$n>1$$ and let $$\mathbb{C}^n$$ denote the complex $$n$$-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings $$F:\mathbb{C}^n\to\mathbb{C}^n$$ and for holomorphic automorphisms of $$\mathbb{C}^n$$ on discrete subsets of $$\mathbb{C}^n$$. We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds into $$\mathbb{C}^n$$. For each closed complex submanifold (or subvariety) $$M\subset\mathbb{C}^n$$ of complex dimension $$m<n$$ we construct a domain $$\Omega\subset \mathbb{C}^n$$ containing $$M$$ and a biholomorphic map $$F:\Omega \to \mathbb{C}^n$$ onto $$\mathbb{C}^n$$ with $$JF\equiv 1$$ such that $$F(M)$$ intersects the image of any nondegenerate entire map $$G:\mathbb{C}^{n-m} \to\mathbb{C}^n$$ at infinitely many points. If $$m=n-1$$, we construct $$F$$ as above such that $$\mathbb{C}^n\setminus F(M)$$ is hyperbolic. In particular, for each $$m\geq 1$$ we construct proper holomorphic embeddings $$F:\mathbb{C}^m \to\mathbb{C}^{m+1}$$ such that the complement $$\mathbb{C}^{m+1} \setminus F(\mathbb{C}^m)$$ is hyperbolic.

##### MSC:
 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 32Q40 Embedding theorems for complex manifolds 32M05 Complex Lie groups, group actions on complex spaces 32Q28 Stein manifolds
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