The standard eigenfunctions ϕ λ = e i⟨λ, x⟩ on flat tori ℝ n /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.
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