Spaces with polynomial hulls that contain no analytic discs
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Mathematische Annalen
Spaces with polynomial hulls that contain no analytic discs Alexander J. Izzo1 Received: 13 March 2018 / Revised: 22 February 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Extensions of the notions of polynomial and rational hull are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is presented. As a consequence it is shown that there exists a Cantor set in C3 with a nontrivial polynomial hull that contains no analytic discs. Using this Cantor set, it is shown that there exist arcs and simple closed curves in C4 with nontrivial polynomial hulls that contain no analytic discs. This answers a question raised by Bercovici in 2014 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 C N , for some N , in such a way as to have a nontrivial polynomial 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. Mathematics Subject Classification 32E20 · 46J10 · 46J15
1 Introduction It was once conjectured that whenever the polynomial hull X of a compact set X in C N is strictly larger than X , the complementary set X \ X must contain an analytic disc. This conjecture was disproved by Stolzenberg [25]. However, when X is a smooth one-dimensional manifold, the set X \ X , if nonempty, is a one-dimensional analytic variety as was also shown by Stolzenberg [26] (strengthening earlier results of several
Communicated by Ngaiming Mok. Dedicated to the memory of Donald Sarason.
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Alexander J. Izzo [email protected] Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, USA
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A. J. Izzo
mathematicians). In contrast, recent work of the author, Samuelsson Kalm, and Wold [18] and of the author and Stout [19] shows that every smooth manifold of dimension X\X strictly greater than one smoothly embeds in some C N as a subspace X such that is nonempty but contains no analytic discs. In response to a talk on the above results given by the author, Hari Bercovici raised the question of whether a nonsmooth one-dimensional manifold can have polynomial hull containing no analytic discs. This question was the motivation for the present paper and will be answered affirmatively. In fact, it will be shown, in Theorem 1.1 below, that every uncountable, compact subspace of a Euclidean space can be embedded in some C N so as to have polynomial hull containing no analytic discs. It is pleasure to thank Bercovici for his question which had a very stimulating effect on the author’s research. A similar question was in fact raised by Wermer [30] more than 60 years ago. Wermer observed that for φ1 and φ2 continuous complex-valued functions separating p
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