Riemannian Manifolds with Uniformly Bounded Eigenfunctions

The standard eigenfunctions φλ = ei〈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 L2normalized eigenfunctions have uniformly bounded L-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. 0. Introduction This paper is concerned with the relation between the dynamics of the geodesic flow G t on the unit sphere bundle SM of a compact Riemannian manifold (M, g) and the growth rate of the L-norms of its L2-normalized 1-eigenfunctions (or “modes”) {φλ}. Let Vλ := {φ : 1φλ = λφλ} denote the λ-eigenspace for λ ∈ Sp(1), and define L(λ, g) = sup φ∈Vλ ||φ|| L2 =1 ||φ||L∞, ` (λ, g) = inf ONB{φ j }∈Vλ ( sup j=1,...,dim Vλ ||φ j ||L∞ ) . (1) The universal bound L(λ, g) = 0(λ(n−1)/4) holds for any (M, g) in consequence of the local Weyl law (see [Ho]):