Duke Mathematical Journal

Riemannian manifolds with uniformly bounded eigenfunctions

John A. Toth and Steve Zelditch

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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.

Article information

Source
Duke Math. J. Volume 111, Number 1 (2002), 97-132.

Dates
First available: 18 June 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1087575008

Mathematical Reviews number (MathSciNet)
MR1876442

Digital Object Identifier
doi:10.1215/S0012-7094-02-11113-2

Zentralblatt MATH identifier
1022.58013

Subjects
Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 53D25: Geodesic flows

Citation

Toth, John A.; Zelditch, Steve. Riemannian manifolds with uniformly bounded eigenfunctions. Duke Mathematical Journal 111 (2002), no. 1, 97--132. doi:10.1215/S0012-7094-02-11113-2. http://projecteuclid.org/euclid.dmj/1087575008.


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