Looping directions and integrals of eigenfunctions over submanifolds

Abstract

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_{\lambda }$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric $g$, i.e \[ -Δ_g e_λ= λ^2 e_λ\qquad \text{ and } \qquad \| e_λ\|_{L^2(M)} = 1. \] Let $\Sigma$ be a $d$-dimensional submanifold and $d\mu$ a smooth, compactly supported measure on $\Sigma$. It is well-known (e.g. proved by Zelditch in far greater generality) that \[ \int_Σe_λ\, dμ= O(λ^\frac{n-d-1}{2}). \] We show this bound improves to $o({\lambda }^{\frac{n-d-1}{2}})$ provided the set of looping directions, \[ \mathcal{L}_Σ = \{ (x,ξ) \in SN^*Σ: Φ_t(x,ξ) \in SN^*Σ\text{ for some } t > 0 \} \] has measure zero as a subset of $S{N}^{\ast }\Sigma$, where here ${\Phi }_t$ is the geodesic flow on the cosphere bundle $S^{\ast }M$ and $S{N}^{\ast }\Sigma$ is the unit conormal bundle over $\Sigma$.