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 $\Delta_g$ be the Laplace-Beltrami operator corresponding to the metric $g$ on $M$, and let $e_\lambda$ be $L^2$-normalized eigenfunctions of $\Delta_g$ with eigenvalue $\lambda$, i.e. [ -\Delta_g e_\lambda = \lambda^2 e_\lambda. ] We prove [ \left| \int_{\mathbb R} b(t) e_\lambda (\gamma(t)) \, dt \right| = o(1) \quad \text{ as } \lambda \to \infty ] where $b$ is a smooth, compactly supported function on $\mathbb R$ and $\gamma$ is a curve parametrized by arc-length whose geodesic curvature $\kappa(\gamma(t))$ avoids two critical curvatures $\mathbf k(\gamma'^\perp(t))$ and $\mathbf k(-\gamma'^{\perp}(t))$ 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 $-v$.