Abstract
In this note we study the systoles of convex hypersurfaces in \({\mathbb {R}}^{2n}\) invariant under an anti-symplectic involution. We investigate a uniform upper bound of the ratio between the systole and the symmetric systole of the hypersurfaces using symplectic capacities from Floer theory. We discuss various concrete examples in which the ratio can be understood explicitly.
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Acknowledgements
The authors cordially thank Urs Frauenfelder, Jungsoo Kang, and Felix Schlenk for fruitful discussions, Yaron Ostrover for helpful comments. They also thank the anonymous referee for valuable suggestions. A part of this work was done while MK visited Korea Institute for Advanced Study. The authors are grateful for its warm hospitality. JK was partially supported by the POSCO Science Fellowship of POSCO TJ Park Foundation SK was supported by the grant 200021-181980/1 of the Swiss National Foundation. MK was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2021R1F1A1060118) and by Sunchon National University Research Fund in 2021 (Grant number: 2021-0238).
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This article is part of the topical collection “Symplectic geometry - A Festschrift in honour of Claude Viterbo’s 60th birthday” edited by Helmut Hofer, Alberto Abbondandolo, Urs Frauenfelder, and Felix Schlenk.
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Kim, J., Kim, S. & Kwon, M. Remarks on the systoles of symmetric convex hypersurfaces and symplectic capacities. J. Fixed Point Theory Appl. 24, 28 (2022). https://doi.org/10.1007/s11784-022-00953-w
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DOI: https://doi.org/10.1007/s11784-022-00953-w