Riemannian manifolds with uniformly bounded eigenfunctions

2002/01/15 by John A. Toth, Steve Zelditch · 3 citations
Physics and Astronomy · Mathematics · #Quantum chaos and dynamical systems #Geometry and complex manifolds #Spectral Theory in Mathematical Physics

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

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

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