Looping Directions and Integrals of Eigenfunctions over Submanifolds

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., -Δgeλ=λ2eλand∥eλ∥L2(M)=1.Let Σ be a d-dimensional submanifold and d μ a smooth, compactly supported measure on Σ. It is well known (e.g., proved by Zelditch, Commun Partial Differ Equ 17(1–2):221–260, 1992 in far greater generality) that ∫Σeλdμ=O(λn-d-12).We show this bound improves to o(λn-d-12) provided the set of looping directions, LΣ={(x,ξ)∈SN∗Σ:Φt(x,ξ)∈SN∗Σfor somet>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.