TY - JOUR

T1 - Short geodesic loops and Lp norms of eigenfunctions on large genus random surfaces

AU - Gilmore, Clifford

AU - Masson, Etienne Le

AU - Sahlsten, Tuomas

AU - Thomas, Joe

PY - 2021/4/5

Y1 - 2021/4/5

N2 - 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 Weil-Petersson volume have almost surely at most one such loop of length less than clogg for small enough c>0. This allows us to deduce that the Lp norms of L2 normalised eigenfunctions on X are bounded by 1/√logg almost surely in the large genus limit for any p>2+ε for ε>0 depending on the spectral gap λ1(X) of X.

AB - 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 Weil-Petersson volume have almost surely at most one such loop of length less than clogg for small enough c>0. This allows us to deduce that the Lp norms of L2 normalised eigenfunctions on X are bounded by 1/√logg almost surely in the large genus limit for any p>2+ε for ε>0 depending on the spectral gap λ1(X) of X.

UR - https://www.research.manchester.ac.uk/portal/en/publications/short-geodesic-loops-and-lpnorms-of-eigenfunctions-on-large-genus-random-surfaces(3e5bcdd8-43f7-42b6-ab00-a134f6a0b693).html

UR - http://www.scopus.com/inward/record.url?scp=85103620689&partnerID=8YFLogxK

U2 - 10.1007/s00039-021-00556-6

DO - 10.1007/s00039-021-00556-6

M3 - Article

VL - 31

SP - 62

EP - 110

JO - GEOMETRIC AND FUNCTIONAL ANALYSIS

JF - GEOMETRIC AND FUNCTIONAL ANALYSIS

SN - 1016-443X

IS - 1

ER -