Short geodesic loops and Lp norms of eigenfunctions on large genus random surfaces

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Gilmore, Clifford
Le Masson, Etienne
Sahlsten, Tuomas
Thomas, Joe
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We give upper bounds for Lp 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 → +∞, we show that random hyperbolic surfaces X with respect to the WeilPetersson 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 Lp norms of L2 normalised eigenfunctions on X are O(1/√log g) with high probability in the large genus limit for any p > 2 +ε for ε > 0 depending on the spectral gap λ1(X) of X, with an implied constant depending on the eigenvalue and the injectivity radius.
Lp , Eigenfunction , Laplacian , Short geodesic loops
Gilmore, C., Le Masson, E., Sahlsten, T. and Thomas, J. (2021) 'Short geodesic loops and Lp norms of eigenfunctions on large genus random surfaces', Geometric and Functional Analysis, 31, pp. 62–110. doi: 10.1007/s00039-021-00556-6
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