Duke Mathematical Journal

Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds

N. Burq, P. Gérard, and N. Tzvetkov

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Abstract

We estimate the Lp-norm (2p+) of the restriction to a curve of the eigenfunctions of the Laplace-Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere, these estimates are sharp. If the curve has nonvanishing geodesic curvature, we can improve our results. All our estimates are shown to be optimal for the sphere. Moreover, we sketch their extension to higher dimensions.

Résumé

On prouve une estimation de la norme Lp (2p+) de la restriction à une courbe des fonctions propres de l'opérateur de Laplace-Beltrami sur une surface riemannienne. Si la courbe est une géodésique de la sphère, on montre que nos estimations sont optimales. En revanche, si la courbe possède une courbure géodésique non nulle, on améliore le résultat. Toutes nos estimées sont optimales sur la sphère. Nous en esquissons par ailleurs des généralisations aux dimensions supérieures

Article information

Source
Duke Math. J., Volume 138, Number 3 (2007), 445-486.

Dates
First available in Project Euclid: 18 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1182180654

Digital Object Identifier
doi:10.1215/S0012-7094-07-13834-1

Mathematical Reviews number (MathSciNet)
MR2322684

Zentralblatt MATH identifier
1131.35053

Subjects
Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Secondary: 35J15: Second-order elliptic equations 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Citation

Burq, N.; Gérard, P.; Tzvetkov, N. Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds. Duke Math. J. 138 (2007), no. 3, 445--486. doi:10.1215/S0012-7094-07-13834-1. https://projecteuclid.org/euclid.dmj/1182180654


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