Eigenfunctions with few critical points.
type: Item
instance of: scholarly article
title: Eigenfunctions with few critical points.
zbMATH Open document ID: 1038.58036
author: Dmitry Jakobson
author: Nikolai Nadirashvili
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: The authors consider critical points of eigenfunctions of the Laplacian on the the two-dimensional torus with a metric of the form \[ g = q(x)(dx^2 + dy^2). \] In particular, they show that there exists a function \(q\) and a sequence of eigenvalues of the Laplacian with respect to \(g\) such that the corresponding eigenvalues converge to infinity but the number of critical points of the eigenfunctions remains bounded. This result is a negative answer of the question raised by \textit{Shing-Tung Yau} [Ann. Math. Stud. 102, 669--706 (1982; Zbl 0479.53001)]. Yau suggested that the number of critical points increases with the growth of the eigenvalue.
reviewed by: Olaf Post
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 34B99
Mathematics Subject Classification ID: 34L20
zbMATH DE Number: 1782620
zbMATH Keywords: Laplacian
zbMATH Keywords: eigenfunction
zbMATH Keywords: critical point
zbMATH Keywords: WKB
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214425450
OpenAlex ID: W1531681815
Wikidata QID: Q115193560
DOI: 10.4310/JDG/1214425450
description: scientific article; zbMATH DE number 1782620