Integrals of eigenfunctions over curves in surfaces of nonpositive curvature

Let (M, g) be a compact, 2-dimensional Riemannian manifold with nonpositive sectional curvature. Let ∆g be the Laplace-Beltrami operator corresponding to the metric g on M, and let e_λ be L²-normalized eigenfunctions of ∆g with eigenvalue λ, i.e. −∆geλ = λ²eλ. We prove (Formula Presented) where b is a smooth, compactly supported function on R and γ is a curve parametrized by arc-length whose geodesic curvature κ(γ(t)) avoids two critical curvatures k(γ^′(t)) and k(−γ^′(t)) for each t ∈ supp b. k(v) denotes the curvature of a circle with center taken to infinity along the geodesic ray in direction −v. Chen and Sogge prove in [2] the same decay for geodesics in M with strictly negative curvature. After performing a standard reduction, they lift the relevant quantity to the universal cover and then use the Hadamard parametrix to reduce the problem to bounding a sum of oscillatory integrals with a geometric phase functions. They use the Gauss-Bonnet theorem to obtain bounds on the Hessian of these phase functions and conclude their argument with stationary phase. Our argument follows theirs, except we prove and use properties of the curvature of geodesic circles to obtain bounds on the Hessian of the phase functions.