Riemannian Manifolds With Uniformly Bounded Eigenfunctions

TL;DR

This paper investigates whether the property of having uniformly bounded eigenfunctions characterizes flat manifolds, providing an affirmative answer for compact Riemannian manifolds with completely integrable geodesic flows.

Contribution

It proves that compact Riemannian manifolds with completely integrable geodesic flows have eigenfunctions with uniformly bounded $L^{inity}$-norms, characterizing flat manifolds among this class.

Findings

Eigenfunctions on flat tori have bounded $L^{inity}$-norms independent of eigenvalues.

On irrational flat tori, eigenfunctions are uniformly bounded in $L^{inity}$-norm.

The property characterizes flat manifolds among compact Riemannian manifolds with integrable geodesic flows.

Abstract

The standard eigenfunctions $\phi_{\lambda} = e^{i < \lambda, x >}$ on flat tori $\R^n / L$ have $L^{\infty}$-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that $L^2$-normalized eigenfunctions have uniformly bounded $L^{\infty}$-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with completely integrable geodesic flows.