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Unbounded Kobayashi Hyperbolic Domains in \({\mathbb {C}}^n\)

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Abstract

We first give a sufficient condition, issued from pluripotential theory, for an unbounded domain in the complex Euclidean space \({\mathbb {C}}^n\) to be Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex domain in \({\mathbb {C}}^3\) that is Kobayashi hyperbolic and has a nonempty core. In particular, this domain is not biholomorphic to a bounded domain in \({\mathbb {C}}^3\) and the mentioned above sufficient condition for Kobayashi hyperbolicity is not necessary.

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Acknowledgements

The authors wish to thank the anonymous referee for precious comments and suggestions, which improved the content of the paper. In particular, Corollary 1 and Example 1 were proposed by the referee. Lemma 2 and its proof were also modified following the referee’s suggestion, as well as different other points all along the paper. Part of this work was done while the second author was a visitor at the Capital Normal University (Beijing). It is his pleasure to thank this institution for its hospitality and good working conditions. The authors also would like to thank Fusheng Deng for his remark related to the definition of the antipeak function which slightly strengthen the statement of Theorem 1.

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Authors and Affiliations

  1. Université Grenoble Alpes, CNRS, IF, 38000, Grenoble, France

    Hervé Gaussier

  2. Department of Mathematics, University of Wuppertal, 42119, Wuppertal, Germany

    Nikolay Shcherbina

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  1. Hervé Gaussier
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  2. Nikolay Shcherbina
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Correspondence to Hervé Gaussier.

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Gaussier, H., Shcherbina, N. Unbounded Kobayashi Hyperbolic Domains in \({\mathbb {C}}^n\). Math. Z. 298, 289–305 (2021). https://doi.org/10.1007/s00209-020-02596-4

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