Spaces with polynomial hulls that contain no analytic discs

2018/01/07 by Izzo, Alexander J.
#32E20 #46J10 #46J15 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.02252

Abstract

Extensions of the notions of polynomially and rationally convex hulls are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomially convex hulls containing no analytic discs is presented. As a consequence it is shown that there exists a Cantor set X in \mathbb C³ with a nontrivial polynomially convex hull that contains no analytic discs. Using this Cantor set, it is shown that there exist arcs and curves in \mathbb C⁴ with nontrivial polynomially convex hulls that contain no analytic discs. This answers a question raised a few years ago by Bercovici and can be regarded as a partial answer to a question raised by Wermer over 60 years ago. More generally, it is shown that every uncountable, compact subspace of a Euclidean space can be embedded as a subspace X of \mathbb C^N, for some N, in such a way as to have a nontrivial polynomially convex hull that contains no analytic discs. In the case when the topological dimension of the space is at most one, X can be chosen so as to have the stronger property that P(X) has a dense set of invertible elements.

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