Riemannian manifolds with uniformly bounded eigenfunctions

The standard eigenfunctions ϕ_λ = e^{i⟨λ, x⟩} on flat tori ℝⁿ/L have L^∞-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that L^∞-normalized eigenfunctions have uniformly bounded L1-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.