Explicit Bounds on Integrals of Eigenfunctions Over Curves in Surfaces of Nonpositive Curvature

Emmett Lyons Wyman*

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

Let (M, g) be a compact Riemannian surface with nonpositive sectional curvature and let γ be a closed geodesic in M. And let eλ be an L2-normalized eigenfunction of the Laplace–Beltrami operator Δ g with - Δ geλ= λ2eλ. Sogge et al. (Camb J Math 5(1):123–151, 2017) showed using the Gauss–Bonnet Theorem that ∫γeλds=O((logλ)-1/2), an improvement over the general O(1) bound. We show this integral enjoys the same decay for a wide variety of curves, where M has nonpositive sectional curvature. These are the curves γ whose geodesic curvature avoids, pointwise, the geodesic curvature of circles of infinite radius tangent to γ.

Original languageEnglish (US)
Pages (from-to)3204-3232
Number of pages29
JournalJournal of Geometric Analysis
Volume30
Issue number3
DOIs
StatePublished - Jul 1 2020

Keywords

  • Eigenfunction restrictions
  • Nonpositive curvature
  • Period integrals

ASJC Scopus subject areas

  • Geometry and Topology

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