Duke Mathematical Journal

Riemannian manifolds with uniformly bounded eigenfunctions

John A. Toth and Steve Zelditch

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 18 June 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

Export citation


  • R. Abraham and J. E. Marsden, Foundations of Mechanics, 2d ed., Benjamin/Cummings, Reading, Mass., 1978.
  • V. I. Arnold, Modes and quasimodes (in Russian), Funktsional. Anal. i Priložen. 6, no. 2 (1972), 12--20.; English translation in Funct. Anal. Appl. 6 (1972), 94--101.
  • M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), 2083--2091.
  • --. --. --. --., Semi-classical mechanics in phase space: A study of Wigner's function, Philos. Trans. Roy. Soc. London Ser. A 287 (1977), 237--271.
  • M. Bialy and L. Polterovich, Hopf-type rigidity for Newton equations, Math. Res. Lett. 2 (1995), 695--700.
  • P. Bleher, D. Kosygin, and Ya. G. Sinaĭ, Distribution of energy levels of quantum free particle on the Liouville surface and trace formulae, Comm. Math. Phys. 179 (1995), 375--403.
  • D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994), 259--269.
  • A.-M. Charbonnel, Comportement semi-classique du spectre conjoint d'opérateurs pseudodifférentiels qui commutent, Asympt. Anal. 1 (1988), 227--261.
  • Y. Colin de Verdière, Quasi-modes sur les variétés riemanniennes, Invent. Math. 43 (1977), 15--52.
  • --. --. --. --., Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques, Comment. Math. Helv. 54 (1979), 508--522.
  • --. --. --. --., Spectre conjoint d'opérateurs pseudo-différentiels qui commutent, II: Le cas intégrable, Math. Z. 171 (1980), 51--73.
  • Y. Colin de Verdière and B. Parisse, Équilibre instable en régime semi-classique, I: Concentration microlocale, Comm. Partial Differential Equations 19 (1994), 1535--1563.
  • C. Croke and B. Kleiner, On tori without conjugate points, Invent. Math. 120 (1995), 241--257.
  • J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687--706.
  • A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verh. Deutsch. Phys. Ges. 19 (1917), 82--92.
  • V. Guillemin and S. Sternberg, Geometric Asymptotics, Math. Surveys 14, Amer. Math. Soc., Providence, 1977.
  • --. --. --. --., Homogeneous quantization and multiplicities of group representations, J. Funct. Anal. 47 (1982), 344--380.
  • G. J. Heckman, Quantum integrability for the Kovalevsky top, Indag. Math. (N.S.) 9 (1998), 359--365.
  • B. Helffer, Semi-classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Math. 1336, Springer, Berlin, 1988.
  • L. Hörmander, The Analysis of Linear Partial Differential Operators, IV: Fourier Integral Operators, Grundlehren. Math. Wiss. 275, Springer, Berlin, 1985.
  • D. Jakobson and S. Zelditch, ``Classical limits of eigenfunctions for some completely integrable systems'' in Emerging Applications of Number Theory (Minneapolis, 1996), IMA Vol. Math. Appl. 109, Springer, New York, 1999.
  • A. Knauf, Closed orbits and converse KAM theory, Nonlinearity 3 (1990), 961--973.
  • D. Kosygin, A. Minasov, and Ya. G. Sinaĭ, Statistical properties of the spectra of Laplace-Beltrami operators on Liouville surfaces (in Russian), Uspekhi Mat. Nauk 48, no. 4 (1993), 3--130.; English translation in Russian Math. Surveys 48, no. 4 (1993), 1--142.
  • F. Lalonde and J.-C. Sikorav, Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Comment. Math. Helv. 66 (1991), 18--33.
  • E. Lerman, A convexity theorem for torus actions on contact manifolds, preprint.
  • --------, Contact toric manifolds, preprint.
  • E. Lerman and N. Shirokova, Toric integrable geodesic flows, preprint.
  • R. Mañé, Ergodic Theory and Differentiable Dynamics, Ergeb. Math. Grenzgeb. (3) 8, Springer, Berlin, 1987.
  • --. --. --. --., ``On a theorem of Klingenberg'' in Dynamical Systems and Bifurcation Theory (Rio de Janeiro, 1985), Pitman Res. Notes Math. Ser. 160, Longman Sci. Tech., Harlow, England, 1987, 319--345.
  • C. D. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth, to appear in Duke Math. J.
  • J. A. Toth, Various quantum mechanical aspects of quadratic forms, J. Funct. Anal. 130 (1995), 1--42.
  • --. --. --. --., Eigenfunction localization in the quantized rigid body, J. Differential Geom. 43 (1996), 844--858.
  • --. --. --. --., On the quantum expected values of integrable metric forms, J. Differential Geom. 52 (1999), 327--374.
  • J. A. Toth and S. Zelditch, (L^p)-estimates of eigenfunctions in the completely integrable case, preprint, 2000.
  • J. M. VanderKam, (L\sp \infty) norms and quantum ergodicity on the sphere, Internat. Math. Res. Notices 1997, 329--347., ; Correction, Internat. Math. Res. Notices 1998, 65.
  • J. A. Wolf, Spaces of Constant Curvature, 5th ed., Publish or Perish, Houston, 1984.
  • J. A. Yorke, Periods of periodic solutions and the Lipschitz constant, Proc. Amer. Math. Soc. 22 (1969), 509--512.
  • S. Zelditch, Quantum transition amplitudes for ergodic and for completely integrable systems, J. Funct. Anal. 94 (1990), 415--436.