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A properly embedded holomorphic disc in the ball with finite area and dense boundary

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In this paper we construct a properly embedded holomorphic disc in the unit ball \(\mathbb {B}^2\) of \(\mathbb {C}^2\) having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on \(\mathbb {B}^2\); on the other hand, its boundary curve is everywhere dense in the sphere \(b\mathbb {B}^2\). A similar result is proved in higher dimensions. Our construction is based on an approximation result in contact geometry, also proved in the paper.

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Acknowledgements

This research was supported in part by the research program P1-0291 and Grant J1-7256 from ARRS, Republic of Slovenia. I wish to thank Filippo Bracci for having brought to my attention the question answered in the paper, Bo Berndtsson for the communication regarding the reference [4], Josip Globevnik for helpful discussions and the reference to his work [12] with E. L. Stout, Finnur Lárusson for remarks concerning the exposition, and Erlend F. Wold for the communication on Sect. 4.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia

    Franc Forstnerič

  2. Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, Slovenia

    Franc Forstnerič

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  1. Franc Forstnerič
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Correspondence to Franc Forstnerič.

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Dedicated to Edgar Lee Stout for his eightieth birthday.

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Forstnerič, F. A properly embedded holomorphic disc in the ball with finite area and dense boundary. Math. Ann. 373, 719–742 (2019). https://doi.org/10.1007/s00208-018-1686-8

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  • DOI: https://doi.org/10.1007/s00208-018-1686-8

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