Integrals of eigenfunctions over curves in surfaces of nonpositive curvature

Abstract

Let $(M,g)$ be a compact, 2-dimensional Riemannian manifold with nonpositive sectional curvature. Let ${\Delta }_g$ be the Laplace-Beltrami operator corresponding to the metric $g$ on $M$, and let $e_{\lambda }$ be $L^2$-normalized eigenfunctions of ${\Delta }_g$ with eigenvalue $\lambda$, i.e. \[ -Δ_g e_λ= λ^2 e_λ. \] We prove \[ \left| \int_{\mathbb R} b(t) e_λ(γ(t)) \, dt \right| = o(1) \quad \text{ as } λ\to \infty \] where $b$ is a smooth, compactly supported function on $\mathbb{R}$ and $\gamma$ is a curve parametrized by arc-length whose geodesic curvature $\kappa (\gamma (t))$ avoids two critical curvatures $\mathbf{k}({\gamma }^{\prime \perp }(t))$ and $\mathbf{k}(-{\gamma }^{\prime \perp }(t))$ for each $t\in \mathrm{supp}b$. $\mathbf{k}(v)$ denotes the curvature of a circle with center taken to infinity along the geodesic ray in direction $-v$.

References in corpus (1)

• On integrals of eigenfunctions over geodesics

Cited by in corpus (3)

• Improved Generalized Periods estimates on Riemannian Surfaces with Nonpositive Curvature
• 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