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The ends of discs. (English) Zbl 0605.32006
We quote the introduction: ”We study the boundary behavior of maps from the unit disc U into $${\mathbb{C}}^ n$$. Our results are of three kinds. First, we show that if $$f: U\to {\mathbb{C}}^ n$$ is a holomorphic map that is injective on U, then $$f$$ is injective on bU, granted that $$f(bU)$$ is a smooth simple closed curve and that $$f$$ is smooth on bU. Some smoothness is necessary, for examples show that continuity by itself does not suffice for the conclusion. Our second set of results has to do with the following question. If $$f: U\to D$$ is a proper holomorphic map, D a bounded, strictly convex domain in $${\mathbb{C}}^ n$$, how do the radial images $$I_{f}(\theta)=\{f(re^{i\theta}): 0<r<1\}$$ approach bD? If $$f'\in H^ 1$$, then almost all of the $$I_{f}(\theta)$$ approach bD nontangentially. Other results of this general kind are obtained. Finally, we have some boundary uniqueness theorems for discs; in particular if $$f$$, g: $$\bar U\to \bar B_ 2$$ are continuous maps holomorphic in U with $$f(bU)$$ and g(bU) rectifiable simple closed curves that lie in $$bB_ 2$$, then either $$f(U)=g(U)$$ or else $$f(bU)\cap g(bU)$$ has zero length.
Our methods are a mixture of elementary geometric considerations and appeal to some sophisticted results from modern geometric function theory.”
Reviewer: E.Straube

##### MSC:
 32E35 Global boundary behavior of holomorphic functions of several complex variables 32H35 Proper holomorphic mappings, finiteness theorems
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