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An embedding of \(\mathbb C\) in \(\mathbb C^2\) with hyperbolic complement. (English) Zbl 0864.32013
The following results are proven. Theorem 4.1. Let \(X\) be a closed 1-dimensional complex subvariety of \(\mathbb C^2\) and \(B\) a closed ball disjoint of \(X\). Then there is a domain \(D\subset \mathbb C^2-B\) containing \(X\) and a biholomorphic map from \(D\) onto \(\mathbb C^2\) such that \(\mathbb C^2-\phi(X)\) is Kobayashi hyperbolic. Moreover, all nonconstant images of \(\mathbb C\) in \(\mathbb C^2\) intersect \((X)\) in infinitely many points.
As consequences one gets:
Theorem 1.1. There is a proper holomorphic embedding \(\phi:\mathbb C \to \mathbb C^2\) such that \(\mathbb C^2-\phi(\mathbb C)\) is Kobayashi hyperbolic.
Theorem 1.2. There exists a proper holomorphic embedding of \(\mathbb C\) into \(\mathbb C^2\) such that any nonconstant holomorphic image of \(\mathbb C\) intersects this embedding infinitely many times.

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
Full Text: DOI
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