On the essential spectrum of the Laplacian on complete manifolds
type: Item
instance of: scholarly article
title: On the essential spectrum of the Laplacian on complete manifolds
zbMATH Open document ID: 0913.58056
author: Hironori Kumura
published in: Journal of the Mathematical Society of Japan
publication date: 1999-05-26T00:00:00Z
review text: The author proves two theorems on when the essential spectrum of the Laplace-Beltrami operator on a complete manifold \(M\) satisfying a generalized ``pole condition'' is equal to an interval \([c,\infty)\). The condition on \(M\) is that it contains an open set \(U\) with compact smooth boundary \(\partial U\) such that the outward normal exponential map is a diffeomorphism. The first theorem says that if the Ricci curvature evaluated in normal directions at distance \(t\) is bounded below by \(-(n-1)\phi(t)\) for some continuous function \(\phi\) vanishing at infinity, then the essential spectrum of \(M\) is equal to \([0,\infty)\). The second theorem states that if \(\Delta r(x)\) converges in a certain sense to a constant \(c\) at infinity, where \(r\) is the distance to \(U\), then the essential spectrum is equal to \([c^2/4,\infty)\).
reviewed by: Mattias Dahl
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35P05
zbMATH DE Number: 1172181
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: essential spectrum
zbMATH Keywords: Ricci curvature
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04910001
OpenAlex ID: W2050807499
DOI: 10.2969/JMSJ/04910001
description: scientific article; zbMATH DE number 1172181