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Visibility of Kobayashi geodesics in convex domains and related properties

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Abstract

Let \(D\subset {\mathbb {C}}^n\) be a bounded domain. A pair of distinct boundary points \(\{p,q\}\) of D has the visibility property provided there exist a compact subset \(K_{p,q}\subset D\) and open neighborhoods \(U_p\) of p and \(U_q\) of q, such that the real geodesics for the Kobayashi metric of D which join points in \(U_p\) and \(U_q\) intersect \(K_{p,q}\). Every Gromov hyperbolic convex domain enjoys the visibility property for any couple of boundary points. The Goldilocks domains introduced by Bharali and Zimmer and the log-type domains of Liu and Wang also enjoy the visibility property. In this paper we relate the growth of the Kobayashi distance near the boundary with visibility and provide new families of convex domains where that property holds. We use the same methods to provide refinements of localization results for the Kobayashi distance, and give a localized sufficient condition for visibility. We also exploit visibility to study the boundary behavior of biholomorphic maps.

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Acknowledgements

We would like to thank Vikramjeet Singh Chandel, Anwoy Maitra, and Amar Deep Sarkar, who pointed out a gap in the proof of Lemma 5.3 in a previous version of the manuscript.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via Della Ricerca Scientifica 1, 00133, Rome, Italy

    Filippo Bracci

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113, Sofia, Bulgaria

    Nikolai Nikolov

  3. Faculty of Information Sciences, State University of Library Studies and Information Technologies, 69A, Shipchenski prohod Str., 1574, Sofia, Bulgaria

    Nikolai Nikolov

  4. Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS UPS, 31062, Toulouse Cedex 9, France

    Pascal J. Thomas

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  1. Filippo Bracci
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Correspondence to Pascal J. Thomas.

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Filippo Bracci was partially supported by PRIN 2017 Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics, Ref: 2017JZ2SW5, by GNSAGA of INdAM and by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

Nikolai Nikolov was partially supported by the National Science Fund, Bulgaria under contract DN 12/2.

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Bracci, F., Nikolov, N. & Thomas, P.J. Visibility of Kobayashi geodesics in convex domains and related properties. Math. Z. 301, 2011–2035 (2022). https://doi.org/10.1007/s00209-022-02978-w

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  • DOI: https://doi.org/10.1007/s00209-022-02978-w

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