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 $\varSigma .$