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Polynomial hulls of arcs and curves


Author: Alexander J. Izzo
Journal: Proc. Amer. Math. Soc. 149 (2021), 199-207
MSC (2010): Primary 32E20; Secondary 32A38, 32E30, 46J10, 46J15
DOI: https://doi.org/10.1090/proc/15138
Published electronically: October 16, 2020
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Abstract: It is shown that there exist arcs and simple closed curves in $ \mathbb{C}^3$ with nontrivial polynomial hulls that contain no analytic discs. It is also shown that in any bounded, connected Runge domain of holomorphy in $ \mathbb{C}^N$ ($ N \geq 2$) there exist polynomially convex arcs and simple closed curves of almost full measure. These results, which strengthen earlier results of the author, are obtained as consequences of a general result about polynomial hulls of arcs and simple closed curves through compact, totally disconnected sets.


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Additional Information

Alexander J. Izzo
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403
Email: aizzo@bgsu.edu

DOI: https://doi.org/10.1090/proc/15138
Keywords: Polynomial convexity, polynomial hull, hull without analytic structure, analytic disc, dense invertibles, arc, simple closed curve, Cantor set, Runge domain
Received by editor(s): December 21, 2019
Received by editor(s) in revised form: March 14, 2020, and April 16, 2020
Published electronically: October 16, 2020
Additional Notes: The author was supported by NSF Grant DMS-1856010.
Communicated by: Harold P. Boas
Article copyright: © Copyright 2020 American Mathematical Society