Advertisement

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
Article
  • 37 Downloads

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 \(\varSigma .\)

Keywords

Submanifolds Eigenfunctions Kuznecov formulae 

Mathematics Subject Classification

58J50 35P20 

Notes

Acknowledgements

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

References

  1. 1.
    Chen, X., Sogge, C.D.: On integrals of eigenfunctions over geodesics. Proc. Am. Math. Soc. 143(1), 151–161 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Helgason, S.: Geometric Analysis on Symmetric Spaces. American Mathematical Society, Providence (2014)zbMATHGoogle Scholar
  3. 3.
    Hezari, H., Riviere, G.: Equidistribution of toral eigenfunctions along hypersurfaces (01 2018)Google Scholar
  4. 4.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. I, 2nd edn. Springer, Berlin (1990)zbMATHGoogle Scholar
  5. 5.
    Sogge, C.D.: Hangzhou Lectures on Eigenfunctions of the Laplacian. Annals of Mathematics Studies, vol. 188. Princeton University Press, Princeton (2014)zbMATHGoogle Scholar
  6. 6.
    Sogge, C.D.: Fourier Integrals in Classical Aanalysis. Cambridge Tracts in Mathematics, 2nd edn, vol 210. Cambridge University Press, Cambridge (2017)Google Scholar
  7. 7.
    Sogge, C.D., Zelditch, S.: Riemannian manifolds with maximal eigenfunction growth. Duke Math. J. 114(3), 387–437 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sogge, C.D., Toth, J.A., Zelditch, S.: About the blowup of quasimodes on Riemannian manifolds. J. Geom. Anal. 21(1), 150–173 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sogge, C.D., Xi, Y., Zhang, C.: Geodesic period integrals of eigenfunctions on Riemannian surfaces and the Gauss–Bonnet theorem. Camb. J. Math. 5(1), 123–151 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Wyman, E.: Explicit bounds on integrals of eigenfunctions over curves in surfaces of nonpositive curvature (preprint, 2017)Google Scholar
  11. 11.
    Wyman, E.: Integrals of eigenfunctions over curves in surfaces of nonpositive curvature (preprint, 2017)Google Scholar
  12. 12.
    Zelditch, S.: Kuznecov sum formulae and Szegő limit formulae on manifolds. Commun. Partial Differ. Equ. 17(1–2), 221–260 (1992)zbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Johns Hopkins UniversityBaltimoreUSA

Personalised recommendations