Two-dimensional slices of nonpseudoconvex domains with rough boundary
For a Riemann domain \(\pi: X\to\mathbb C^n\), \(n\geq3\), let \(E\) be the set of all \(z\in\mathbb C^n\) such that for every two-dimensional complex affine subspace \(P\subset\mathbb C^n\) with \(z\in P\) the Riemann region \(X\cap\pi^{-1}(P)\) is pseudoconvex. The main result of the paper states that for a nonpseudoconvex Riemann domain \(X\) the set \(E\) is contained in a two-codimensional affine complex subspace. It is known that the result is sharp [\textit{N. Nikolov} and \textit{P. J. Thomas}, Indiana Univ. Math. J. 61, No. 3, 1313--1323 (2012; Zbl 1280.32005)]. In the case where \(X=D\subset\mathbb C^n\) and \(\partial D\) is sufficiently smooth, some characterizations of the set \(E\) were previously obtained in the paper [\textit{N. Nikolov} and \textit{P. Pflug}, Math. Z. 272, No. 1--2, 381--388 (2012; Zbl 1255.32006)].
scientific article; zbMATH DE number 6357837