Improved Generalized Periods estimates on Riemannian Surfaces with Nonpositive Curvature

We show that on compact Riemann surfaces of negative curvature, the generalized periods, i.e. the $\nu$-th order Fourier coefficient of eigenfunctions $e_\lambda$ over a period geodesic $\gamma$ goes to 0 at the rate of $O((\log\lambda)^{-1/2})$, if $0<\nu<c_0\lambda$, given any $0<c_0<1$. No such result is possible for the sphere $S^2$ or the flat torus $\mathbb T^2$. Combined with the quantum ergodic restriction result of Toth and Zelditch, our results imply that for a generic closed geodesic $\gamma$ on a compact hyperbolic surface, the restriction $e_{\lambda_j}|_\gamma$ of an orthonormal basis $\{e_{\lambda_j}\}$ has a full density subsequence that goes to zero weakly in $L^2(\gamma)$. Our proof consists of a further refinement of a recent paper by Sogge, Xi and Zhang on the geodesic period integrals ($\nu=0$), which featured the Gauss-Bonnet Theorem as a key quantitative tool to avoid geodesic rectangles on the universal cover of $M$. In contrast, we shall employ the Gauss-Bonnet Theorem to quantitatively avoid geodesic parallelograms. The use of Gauss-Bonnet also enables us to weaken our curvature condition, by allowing the curvature to vanish at an averaged rate of finite type.