zbMATH — the first resource for mathematics

On the polynomial hull of a graph. (English) Zbl 0798.32026
Let \(G \subset \mathbb{R}^ 3\) be a bounded strictly convex domain and let \(\varphi : \partial G \to \mathbb{R}\) be a continuous function. Put \(\Gamma : = \{(z,w) \in \mathbb{C}^ 2:(z, \text{Re} w) \in G\), \(\text{Im} w = \varphi (z, \text{Re} w)\}\) and let \(\widehat \Gamma\) denote the polynomial hull of \(\Gamma\).
The author studies the structure of the set \(\widehat \Gamma \backslash \Gamma\). The main result is the following theorem: There exists a family \((D_ \alpha)_{\alpha \in A} \subset \mathbb{C}^ 2\) of pairwise disjoint holomorphic discs such that \(\widehat \Gamma \backslash \Gamma = \bigcup_{\alpha \in A}D_ \alpha\) and for each \(\alpha \in A\) there exist a simply connected domain \(\Omega_ \alpha \subset \mathbb{C}\) and a function \(f_ \alpha : \overline \Omega_ \alpha \to \mathbb{C}\) with the following properties:
\(f_ \alpha \in {\mathcal C} (\overline \Omega_ \alpha) \cap {\mathcal O} (\Omega_ \alpha)\),
\(D_ \alpha = \{(z,f_ \alpha (z)) : z \in \Omega_ \alpha\}\),
\(\Gamma \supset \{(z,f_ \alpha (z)) : z \in \partial \Omega_ \alpha\}\),
\(\mathbb{C} \backslash \overline \Omega_ \alpha\) is connected,
if the set \(\partial \Omega_ \alpha \backslash \partial \overline \Omega_ \alpha\) is nonempty, then it has a noncountable number of connected components.

32S65 Singularities of holomorphic vector fields and foliations
Full Text: DOI