## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1302-1319 |

Number of pages | 18 |

Journal | Journal of Geometric Analysis |

Volume | 29 |

Issue number | 2 |

DOIs | |

State | Published - Apr 15 2019 |

## Keywords

- Eigenfunctions
- Kuznecov formulae
- Submanifolds

## ASJC Scopus subject areas

- Geometry and Topology