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 periodic 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 \({\mathbb {T}}^2\) . Our proof consists of a further refinement of a 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, one key new observation is that we can use the Gauss–Bonnet Theorem to quantitatively avoid geodesic parallelograms and isosceles trapezoids as well.
