Riemannian manifolds with uniformly bounded eigenfunctions
Toth, John A. ; Zelditch, Steve
Duke Math. J., Tome 115 (2002) no. 1, p. 97-132 / Harvested from Project Euclid
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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.
Publié le : 2002-01-15
Classification:  58J50,  53D25 
@article{1087575008,
     author = {Toth, John A. and Zelditch, Steve},
     title = {Riemannian manifolds with uniformly bounded eigenfunctions},
     journal = {Duke Math. J.},
     volume = {115},
     number = {1},
     year = {2002},
     pages = { 97-132},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1087575008}
}

Toth, John A.; Zelditch, Steve. Riemannian manifolds with uniformly bounded eigenfunctions. Duke Math. J., Tome 115 (2002) no. 1, pp.  97-132. http://gdmltest.u-ga.fr/item/1087575008/