Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds
type: Item
instance of: scholarly article
title: Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds
zbMATH Open document ID: 1131.35053
DOI: 10.1215/S0012-7094-07-13834-1
published in: Duke Mathematical Journal
publication date: 2007-07-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0506394
review text: Let \((M,g)\) be a compact smooth Riemannian manifold of the dimension \(d\) without boundary and \(\Delta\) the associated with the metric \(g\) Laplace-Beltrami operator. Let \(\{ \varphi_\lambda \}\), \(\lambda \geq 0\) be the eigenfunctions of \(\Delta\), i.e. \(-\Delta\varphi_\lambda=\lambda^2\varphi_\lambda\). The authors study the possible growth of the \(L^p\)-norm, \(2 \leq p \leq +\infty\) of the restrictions of \(\varphi_\lambda\) to submanifolds \(\Sigma\) of \(M\) in the simplest case when \(\Sigma\) is a smooth curve \(\gamma :[a,b] \to M\) parametrized by arc length. When the curve is a geodetic it is shown that on the sphere the obtained estimates are sharp, optimal for the sphere.
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 5173335
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: restrictions of eigenfunctions to a curve
zbMATH Keywords: \(L^p\) estimates
author: Nicolas Burq
author: Patrick Gérard
author: Nikolay Tzvetkov
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
arXiv ID: math/0506394
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OpenAlex ID: W2028478844
description: scientific article