2017/06/21 by Wyman, Emmett L. · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1706.06717
Let (M,g) be a compact n-dimensional Riemannian manifold without boundary and eλ be an L2-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric g, i.e -Δg eλ= λ2 eλ and ‖ eλ‖L2(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 ∫Σeλ dμ= O(λ^(n-d-1)/(2)). We show this bound improves to o(λ^(n-d-1)/(2)) provided the set of looping directions, LΣ = \ (x,ξ) ∈ SN^*Σ: Φt(x,ξ) ∈ SN^*Σ for some t gt; 0 \ has measure zero as a subset of SN^*Σ, where here Φt is the geodesic flow on the cosphere bundle S^*M and SN^*Σ is the unit conormal bundle over Σ.