Explicit bounds on integrals of eigenfunctions over curves in surfaces of nonpositive curvature

Abstract

Let $(M,g)$ be a compact Riemannian surface with nonpositive sectional curvature and let $γ$ be a closed geodesic in $M$. And let $e_λ$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator $Δ_g$ with $-Δ_g e_λ= λ^2 e_λ$. Sogge, Xi, and Zhang showed using the Gauss-Bonnet theorem that $$ \int_γe_λ\, ds = O((\logλ)^{-1/2}),$$ an improvement over the general $O(1)$ bound. We show this integral enjoys the same decay for a wide variety of curves, where $M$ has nonpositive sectional curvature. These are the curves $γ$ whose geodesic curvature avoids, pointwise, the geodesic curvature of circles of infinite radius tangent to $γ$.