Improved Generalized Periods estimates on Riemannian Surfaces with Nonpositive Curvature

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.

19 pages, 4 figures, added two corollaries on weak $L^2$ convergence. Our proof generalizes the argument in arXiv:1604.03189 by Sogge-Xi-Zhang to the generalized periods case

References in corpus (2)

• Integrals of eigenfunctions over curves in surfaces of nonpositive curvature
• Inner product of eigenfunctions over curves and generalized periods for compact Riemannian surfaces

Cited by in corpus (3)

• Inner product of eigenfunctions over curves and generalized periods for compact Riemannian surfaces
• Improved Generalized Periods Estimates Over Curves on Riemannian Surfaces with Nonpositive Curvature
• Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions