Integrals of eigenfunctions over curves in surfaces of nonpositive curvature

Abstract

Let $(M,g)$ be a compact, 2-dimensional Riemannian manifold with nonpositive sectional curvature. Let $Δ_g$ be the Laplace-Beltrami operator corresponding to the metric $g$ on $M$, and let $e_λ$ be $L^2$-normalized eigenfunctions of $Δ_g$ with eigenvalue $λ$, i.e. \[ -Δ_g e_λ= λ^2 e_λ. \] We prove \[ \left| \int_{\mathbb R} b(t) e_λ(γ(t)) \, dt \right| = o(1) \quad \text{ as } λ\to \infty \] where $b$ is a smooth, compactly supported function on $\mathbb R$ and $γ$ is a curve parametrized by arc-length whose geodesic curvature $κ(γ(t))$ avoids two critical curvatures $\mathbf k(γ'^\perp(t))$ and $\mathbf k(-γ'^{\perp}(t))$ for each $t \in \operatorname{supp} b$. $\mathbf k(v)$ denotes the curvature of a circle with center taken to infinity along the geodesic ray in direction $-v$.