Riemannian manifolds with uniformly bounded eigenfunctions
John A. Toth,Steve Zelditch +1 more
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TLDR
The standard eigenfunctions of Riemannian flat tori with quantum completely integrable Laplacians have uniformly bounded $L 2 -norms as mentioned in this paper.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.read more
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