Riemannian Manifolds With Uniformly Bounded Eigenfunctions

Details Riemannian Manifolds With Uniformly Bounded Eigenfunctions Riemannian Manifolds With Uniformly Bounded Eigenfunctions

2000-02-15

arxiv.org/abs/math-ph/0002038v2

Physics

Mathematical Physics

Substantially revised. Duke Math Journal, to appear

Scientific paper

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.

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