Geodesic period integrals of eigenfunctions on Riemannian surfaces and the Gauss–Bonnet Theorem

Abstract

We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemannian surfaces of negative curvature period integrals of eigenfunctions e λ over geodesics go to zero at the rate of O((log λ) −1/2 ) if λ are their frequencies. As discussed in [4], no such result is possible in the constant curvature case if the curvature is ≥ 0. Notwithstanding, we also show that these bounds for period integrals are valid provided that integrals of the curvature over all geodesic balls of radius r ≤ 1 are pinched from above by −δr N for some fixed N and δ > 0. This allows, for instance, the curvature to be nonpositive and to vanish of finite order at a finite number of isolated points. Naturally, the above results also hold for the appropriate type of quasi-modes.

Inner Product of Eigenfunctions over Curves and Generalized Periods for Compact Riemannian Surfaces

“…It is shown by Reznikov [Rez15] that on hyperbolic surfaces, if γ is a periodic geodesic or a geodesic circle, the ν-th order Fourier coefficients of e λ | γ is uniformly bounded if ν ≤ c γ λ for some constant c γ depending on γ. Another key insight, which is provided by the results in [SXZ17], is that the first ν Fourier coefficients of e λ (γ(t)) as a function on S 1 , are uniformly bounded by C ν (log λ) − 1 2 . In this spirit, as a byproduct of our argument, we generalize the result of Reznikov to any smooth closed curve over general Riemannian surfaces.…”

“…This allowed the authors to obtain o(1) decay for period integrals using a stationary phase argument involving reproducing kernels for the eigenfunctions. In a recent paper of Sogge, the author and Zhang [SXZ17], this method was further refined, and they managed to show that…”

Improved Critical Eigenfunction Restriction Estimates on Riemannian Surfaces with Nonpositive Curvature

“…The last ingredient of our proof is the following stationary phase lemma developed in [SXZ17], which is a more flexible version of the standard stationary phase arguments. Note that as in [SXZ17] we are not seeking for the optimal power gain in the following lemma, since any power gain would suffice. Suppose further that for 0 ≤ j ≤ N = 4σ −1 (4.2) |∂ j t a(t)| ≤ C j λ j/2 and that…”

“…The proof of Theorem 1.5 will follow the general scheme introduced in [CS15] and [SXZ17] rather closely. In [SXZ17], Gauss-Bonnet Theorem was used to exploit the defects of geodesic quadrilaterals that arise in these arguments.…”

“…a pleasure for the author to thank his collaborators Sogge and Zhang, since this paper would not have been possible without their joint work[SXZ17]. The author also would like to thank Professor Zelditch for suggesting this problem.1 In the sense of[TZ13] …”