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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.

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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