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

## TL;DR

This paper establishes decay bounds for integrals of Laplace eigenfunctions over various curves in nonpositively curved surfaces, extending previous results from closed geodesics to a broader class of curves.

## Contribution

It generalizes decay bounds for eigenfunction integrals from closed geodesics to a wider class of curves in nonpositive curvature surfaces.

## Key findings

- Integral bounds decay as (log λ)^{-1/2} for the curves considered.
- Results apply to curves with geodesic curvature avoiding that of tangent circles.
- Extends previous bounds from geodesics to more general curves.

## Abstract

Let $(M,g)$ be a compact Riemannian surface with nonpositive sectional curvature and let $\gamma$ be a closed geodesic in $M$. And let $e_\lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator $\Delta_g$ with $-\Delta_g e_\lambda = \lambda^2 e_\lambda$. Sogge, Xi, and Zhang showed using the Gauss-Bonnet theorem that $$ \int_\gamma e_\lambda \, ds = O((\log\lambda)^{-1/2}),$$ 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 $\gamma$ whose geodesic curvature avoids, pointwise, the geodesic curvature of circles of infinite radius tangent to $\gamma$.