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Holomorphic embeddings of planar domains in $$\mathbb{C}^ 2$$. (English) Zbl 0847.32030
This is a very interesting embedding result. The authors prove that every bounded finitely-connected domain in $$\mathbb{C}$$ with no isolated boundary points can be properly holomorphically embedded in $$\mathbb{C}^2$$. The authors motivate their construction by describing a procedure which yields an embedding of the unit disc $$\Delta$$ into $$\mathbb{C}^2$$ (it was previously known that $$\Delta$$ and the annulus admit such embeddings). In this case they produce a sequence of maps which are compositions of polynomial shears (considered as defined on domains which are small perturbations of $$\Delta)$$, and which converge to a map which gives a proper holomorphic embedding of a small perturbation of $$\Delta$$ into $$\mathbb{C}^2$$. Of course this perturbed domain is biholomorphic to $$\Delta$$ by the Riemann mapping theorem. In the $$M$$-connected case one needs only consider the case of domains bounded by $$M$$ circles. The proof in this case consists of a more subtle application of perturbation techniques.
Reviewer: I.Graham (Toronto)

##### MSC:
 32H35 Proper holomorphic mappings, finiteness theorems
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##### References:
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