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Analytic discs with rectifiable simple closed curves as ends. (English) Zbl 0644.32003
The following two theorems are proved: a) Let $$\Gamma$$ be a rectifiable simple closed curve in $${\mathbb{C}}^ N$$, $$\Delta$$ the open unit disc in $${\mathbb{C}}$$. If $$f: \Delta\to {\mathbb{C}}^ N-\Gamma$$ is a bounded proper holomorphic map, then f’ belongs to the Hardy space $$H^ 1$$. Moreover f carries $$b\Delta$$ onto $$\Gamma$$ as a covering map of order equal to the multiplicity of the map f.
b) Let $$\Gamma$$ be a compact set of finite length in $${\mathbb{C}}^ N$$ such that, outside a closed subset of zero length, $$\Gamma$$ has a structure of an arc. Suppose that $$f: \Delta\to {\mathbb{C}}^ N-\Gamma$$ is a bounded proper holomorphic map. Then f’ belongs to $$H^ 1.$$
The proofs depend on an extension result concerning bounded holomorphic functions in $$\Delta$$ recently obtained (independently) by Pommerenke and Alexander.
Reviewer: N.Mihalache

##### MSC:
 32B15 Analytic subsets of affine space 32A35 $$H^p$$-spaces, Nevanlinna spaces of functions in several complex variables 32A40 Boundary behavior of holomorphic functions of several complex variables 30E25 Boundary value problems in the complex plane
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