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Existence of Geodesic Spirals for the Kobayashi–Fuks Metric on Planar Domains

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Abstract

In this note, we discuss the following problem: Given a smoothly bounded strongly pseudoconvex domain D in \(\mathbb {C}^n\), can we guarantee the existence of geodesics for the Kobayashi–Fuks metric which “spiral around” in the interior of D? We find an affirmative answer to the above question for \(n=1\) when D is not simply connected.

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Acknowledgements

The author would like to thank D. Borah for suggesting the problem, as well as for his constant encouragement and guidance. The author is also grateful to Prof. G. Herbort for his kind inputs. Some of the results presented here, especially in Sect. 2, has benefited from the interactions the author had with Prof. Herbort over emails. Finally, the author thanks the anonymous referee for valuable suggestions for improving the exposition herein, especially for pointing out the result that is presented in Corollary 1.4.

Funding

The research of the author was supported by the Institute Post-doctoral Fellowship program at Indian Institute of Technology Bombay (Institute ID: 20002836).

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  1. Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, 400076, India

    Debaprasanna Kar

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Communicated by Irene Sabadin.

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This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, Michael Shapiro and Daniele Struppa.

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Kar, D. Existence of Geodesic Spirals for the Kobayashi–Fuks Metric on Planar Domains. Complex Anal. Oper. Theory 17, 46 (2023). https://doi.org/10.1007/s11785-023-01355-7

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