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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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