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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.

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