The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1302–1319 | Cite as

Looping Directions and Integrals of Eigenfunctions over Submanifolds

  • Emmett L. WymanEmail author


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 \(\varSigma .\)


Submanifolds Eigenfunctions Kuznecov formulae 

Mathematics Subject Classification

58J50 35P20 



The author would like to thank Yakun Xi for pointing out an error in an earlier draft of this paper.


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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Johns Hopkins UniversityBaltimoreUSA

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