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Short geodesic loops and \(L^p\) norms of eigenfunctions on large genus random surfaces

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Abstract

We give upper bounds for \(L^p\) norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus \(g \rightarrow +\infty \), we show that random hyperbolic surfaces X with respect to the Weil-Petersson volume have with high probability at most one such loop of length less than \(c \log g\) for small enough \(c > 0\). This allows us to deduce that the \(L^p\) norms of \(L^2\) normalised eigenfunctions on X are \(O(1/\sqrt{\log g})\) with high probability in the large genus limit for any \(p > 2 + \varepsilon \) for \(\varepsilon > 0\) depending on the spectral gap \(\lambda _1(X)\) of X, with an implied constant depending on the eigenvalue and the injectivity radius.

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Notes

  1. Note that by the Gauss-Bonnet theorem the genus g and the volume of a compact hyperbolic surface |X| are related by the formula \(|X| = 2\pi (2g-2),\) and are therefore equivalent parameters in this context.

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Acknowledgements

We thank Bram Petri for patiently explaining to us the details of [MP19] and sharing his ideas on the proof of Theorem 1.3. We are grateful to Ara Basmajian, Mikolaj Fraczyk, Ilmari Kangasniemi, Vadim Kulikov, Michael Magee, Laura Monk, Mark Pollicott, Peter Sarnak, Alex Wright and Peter Zograf for many helpful discussions and comments. Finally we thank the anonymous referee for numerous comments that led to an improvement of the presentation and the proof.

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Authors and Affiliations

  1. School of Mathematical Sciences, University College Cork, Cork, Ireland

    Clifford Gilmore

  2. Laboratoire de mathématiques AGM, UMR CNRS 8088, CY Cergy-Paris Université, Cergy, France

    Etienne Le Masson

  3. Department of Mathematics, University of Manchester, Manchester, UK

    Tuomas Sahlsten & Joe Thomas

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  1. Clifford Gilmore
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  2. Etienne Le Masson
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  3. Tuomas Sahlsten
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  4. Joe Thomas
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Contributions

The project was conceptualised by E. Le Masson and joined in Summer 2018 by T. Sahlsten, C. Gilmore and J. Thomas. C. Gilmore contributed to discussions on the spectral theory side in Autumn 2018 and then the other three authors, with a significant contribution of J. Thomas, completed the work towards finishing the spectral theory side and adding the random surface component during Summer and Autumn 2019.

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Correspondence to Etienne Le Masson.

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C. Gilmore was supported by the Magnus Ehrnrooth Foundation and the Irish Research Council via a Government of Ireland Postdoctoral Fellowship. E. Le Masson was supported by Initiative d’Excellence Paris-Seine.

T. Sahlsten was supported by a start-up grant from MIMS, University of Manchester. J. Thomas was supported by the Dean’s Award from the University of Manchester.

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Gilmore, C., Le Masson, E., Sahlsten, T. et al. Short geodesic loops and \(L^p\) norms of eigenfunctions on large genus random surfaces. Geom. Funct. Anal. 31, 62–110 (2021). https://doi.org/10.1007/s00039-021-00556-6

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  • DOI: https://doi.org/10.1007/s00039-021-00556-6

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