A nonpluripolar hull with no analytic structure. (English) Zbl 0949.32006

Fornæss, John Erik; Levenberg, Norman

Rassias, Themistocles M. (ed.), Complex analysis in several variables. Palm Harbor, FL: Hadronic Press. 65-74 (1999).

Examples of compact sets \(X\subset {\mathbb C}^n\) such that \(\widehat{X}\backslash X\) (\(\widehat{X}\) the polynomially convex hull of \(X\)) has no analytic structure have been given by G. Stolzenberg [J. Math. Mech. 12, 103-111 (1963; Zbl 0113.29101)] and J. Wermer [Ark. Mat. 20, 129-135 (1982; Zbl 0491.32013)].
The paper under review presents compact sets \(X\subset \partial D\) (\(D\) an arbitrary bounded domain in \({\mathbb C}^2\), \(\widehat{\overline{D}}= \overline{D}\)) with \(\widehat{X}\backslash X\) nonpluripolar, i.e. \(\widehat{X}\backslash X\) is thick in the sense of the pluripotential theory, but it contains no analytic disc.
Applications to maximal plurisubharmonic and extremal plurisubharmonic functions are given.
For the entire collection see [Zbl 0933.00013].

Reviewer: Peter Pflug (Oldenburg)

MSC:

32E20	Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
32V15	CR manifolds as boundaries of domains

Keywords:

polynomial convexity; nonpluripolar sets

Citations:

Zbl 0113.29101; Zbl 0491.32013