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Minimal hulls of compact sets in $ \mathbb{R}^3$


Authors: Barbara Drinovec Drnovšek and Franc Forstnerič
Journal: Trans. Amer. Math. Soc. 368 (2016), 7477-7506
MSC (2010): Primary 53A10, 32U05; Secondary 32C30, 32E20, 49Q05, 49Q15
DOI: https://doi.org/10.1090/tran/6777
Published electronically: December 14, 2015
MathSciNet review: 3471098
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Abstract: The main result of this paper is a characterization of the minimal surface hull of a compact set $ K$ in $ \mathbb{R}^3$ by sequences of conformal minimal discs whose boundaries converge to $ K$ in the measure theoretic sense, and also by $ 2$-dimensional minimal currents which are limits of Green currents supported by conformal minimal discs. Analogous results are obtained for the null hull of a compact subset of $ \mathbb{C}^3$. We also prove a null hull analogue of the Alexander-Stolzenberg-Wermer theorem on polynomial hulls of compact sets of finite linear measure, and a polynomial hull version of classical Bochner's tube theorem.


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Additional Information

Barbara Drinovec Drnovšek
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana – and – Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
Email: barbara.drinovec@fmf.uni-lj.si

Franc Forstnerič
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana – and – Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
Email: franc.forstneric@fmf.uni-lj.si

DOI: https://doi.org/10.1090/tran/6777
Keywords: Minimal surfaces, minimal hulls, holomorphic null curves, null hulls, plurisubharmonic functions, polynomial hulls, positive currents
Received by editor(s): October 24, 2014
Published electronically: December 14, 2015
Article copyright: © Copyright 2015 American Mathematical Society