Improved Generalized Periods estimates on Riemannian Surfaces with Nonpositive Curvature

TL;DR

This paper improves estimates on the decay of generalized Fourier coefficients of eigenfunctions on negatively curved Riemann surfaces, revealing new asymptotic behavior and extending previous results through refined geometric analysis.

Contribution

It provides sharper decay rates for generalized periods on hyperbolic surfaces and introduces a novel use of the Gauss-Bonnet theorem to handle curvature conditions.

Findings

Generalized periods decay at rate O((log λ)^(-1/2)) for negative curvature surfaces.

Results imply restriction of eigenfunctions to geodesics weakly tend to zero.

Curvature conditions can be relaxed to allow vanishing curvature at finite type rates.

Abstract

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.