Riemannian manifolds with uniformly bounded eigenfunctions

Abstract

The standard eigenfunctions $\phi_\lambda=e^{i\langle\lambda,x\rangle}$ on flat tori $\mathbb {R}^n/L$ 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 $^\infty$-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with quantum completely integrable Laplacians.