Looping Directions and Integrals of Eigenfunctions over Submanifolds

Emmett Lyons Wyman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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 languageEnglish (US)
Pages (from-to)1302-1319
Number of pages18
JournalJournal of Geometric Analysis
Volume29
Issue number2
DOIs
StatePublished - Apr 15 2019

Keywords

  • Eigenfunctions
  • Kuznecov formulae
  • Submanifolds

ASJC Scopus subject areas

  • Geometry and Topology

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