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., $$\begin{aligned} -\Delta _g e_\lambda = \lambda ^2 e_\lambda \quad \text {and} \quad \left\| e_\lambda \right\| _{L^2(M)} = 1. \end{aligned}$$ Let $$\varSigma $$ be a d-dimensional submanifold and $$\mathrm{d}\mu $$ a smooth, compactly supported measure on $$\varSigma .$$ It is well known (e.g., proved by Zelditch, Commun Partial Differ Equ 17(1–2):221–260, 1992 in far greater generality) that $$\begin{aligned} \int _\varSigma e_\lambda \, \mathrm{d}\mu = O\left( \lambda ^\frac{n-d-1}{2}\right) . \end{aligned}$$ We show this bound improves to $$o\left( \lambda ^\frac{n-d-1}{2}\right) $$ provided the set of looping directions, $$\begin{aligned} {{\mathcal {L}}}_{\varSigma } = \{ (x,\,\xi ) \in \mathrm{SN}^*\varSigma : \varPhi _t(x,\,\xi ) \in \mathrm{SN}^*\varSigma \text { for some } t > 0 \} \end{aligned}$$ has measure zero as a subset of $$\mathrm{SN}^*\varSigma ,$$ where here $$\varPhi _t$$ is the geodesic flow on the cosphere bundle $$S^*M$$ and $$\mathrm{SN}^*\varSigma $$ is the unit conormal bundle over $$\varSigma .$$