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Riemannian Manifolds With Uniformly Bounded Eigenfunctions

John Toth·Steve Zelditch
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Abstract

The standard eigenfunctions $ϕ_λ = e^{i < λ, x >}$ 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.

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John Toth, Steve Zelditch. 2001-02-05. Riemannian Manifolds With Uniformly Bounded Eigenfunctions. https://arxiv.org/abs/math-ph/0002038

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