Riemannian manifolds with uniformly bounded eigenfunctions

2002/01/15 by John A. Toth, Steve Zelditch · 66 citations
Mathematics · Physics and Astronomy · #Bounded function #Eigenfunction #Eigenvalues and eigenvectors #Geometry #Geometry and complex manifolds #Integrable system #Lambda #Mathematical analysis #Mathematics #Physics #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Spectral Theory in Mathematical Physics #Torus

paper · doi:10.1215/s0012-7094-02-11113-2

published in Duke Mathematical Journal 111(1) (Duke University Press)

openalex publication_date 2002/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15

Abstract

The standard eigenfunctions φ_λ=e^i⟨λ,x⟩ on flat tori \mathbb Rⁿ/L have L^∞-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that L²-normalized eigenfunctions have uniformly bounded ^∞-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with quantum completely integrable Laplacians.

Citations

• Riemannian manifolds with maximal eigenfunction growth
• On global action‐angle coordinates

Cited by

• On the L^∞ norms of spectral projectors on shrinking intervals: the cases of some spheres of revolution and of the Euclidean disk
• Geodesic flows and contact toric manifolds
• Toric integrable geodesic flows in odd dimensions
• Unlabeled Data Help in Graph-Based Semi-Supervised Learning: A Bayesian Nonparametrics Perspective
• Large values of cusp forms on GL(n)
• L 2-Restriction Bounds for Eigenfunctions Along Curves in the Quantum Completely Integrable Case

Related

• Norms of Modes and Quasi-modes revisited
• On the vanishing of eigenfunctions of the Laplacian on tori
• Localized L^p-estimates for eigenfunctions: II
• Concerning the L⁴ norms of typical eigenfunctions on compact surfaces
• Localization properties of high energy eigenfunctions on flat tori