Polynomial hulls with disk fibers over the ball in \(\mathbb{C}^2\)
There are several papers describing the polynomially convex hull of a compact set \(Y \subset {\mathbb{C}}^2\) fibered over the unit circle. The paper under review deals with a compact set \(Y\subset {\mathbb{C}}^3\) fibered over the boundary of the unit ball \(B_2 \subset {\mathbb{C}}^2\). If all fibers \(Y_\lambda\), \(\lambda\in \partial B_2\), are closed discs, i.e. if \(Y = \{ (z,w) \in \partial B_2 \times {\mathbb{C}} : | w-\alpha(z)| \leq 1\}\), if \((B\times{\mathbb{C}}) \setminus \widehat{Y}\) is pseudoconvex, and if \(\#\widehat{Y}_{(0,0)}\geq 2\), then it is shown that \(\widehat{Y} \cap (B_2\times {\mathbb{C}})\) is the union of the analytic graphs over \(B_2\) whose boundaries lie in \(Y\). The proof is based on the analogous result in \({\mathbb{C}}^2\) [cf. \textit{H. Alexander} and \textit{J. Wermer}, Math. Ann. 271, 99-109 (1985; Zbl 0553.32010)] via a slicing argument.
scientific article; zbMATH DE number 1127767