Polynomial hulls with disk fibers over the ball in \(\mathbb{C}^2\)
type: Item
instance of: scholarly article
title: Polynomial hulls with disk fibers over the ball in \(\mathbb{C}^2\)
zbMATH Open document ID: 0898.32008
author: Marshall A. Whittlesey
published in: Michigan Mathematical Journal
publication date: 1998-10-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32E20
zbMATH DE Number: 1127767
reviewed by: Peter Pflug
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005783
OpenAlex ID: W2053899085
DOI: 10.1307/MMJ/1029005783
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description: scientific article; zbMATH DE number 1127767