Looping directions and integrals of eigenfunctions over submanifolds

Abstract

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_λ$ 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 $Σ$ be a $d$-dimensional submanifold and $dμ$ a smooth, compactly supported measure on $Σ$. 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 $o(λ^\frac{n-d-1}{2})$ 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 $SN^*Σ$, where here $Φ_t$ is the geodesic flow on the cosphere bundle $S^*M$ and $SN^*Σ$ is the unit conormal bundle over $Σ$.