Explicit bounds on integrals of eigenfunctions over curves in surfaces of nonpositive curvature

2017/05/04 by Wyman, Emmett L. · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.01688

Abstract

Let (M,g) be a compact Riemannian surface with nonpositive sectional curvature and let γ be a closed geodesic in M. And let e_λ be an L²-normalized eigenfunction of the Laplace-Beltrami operator Δ_g with -Δ_g e_λ= λ² e_λ. Sogge, Xi, and Zhang showed using the Gauss-Bonnet theorem that ∫_γe_λ ds = O((logλ)^-1/2), an improvement over the general O(1) bound. We show this integral enjoys the same decay for a wide variety of curves, where M has nonpositive sectional curvature. These are the curves γ whose geodesic curvature avoids, pointwise, the geodesic curvature of circles of infinite radius tangent to γ.

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