On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds
type: Item
instance of: scholarly article
title: On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds
zbMATH Open document ID: 1269.58011
author: Andrea Cianchi
author: V. Maz'ya
published in: Journal of Differential Geometry
publication date: 2011-10-21T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jdg/1312998232
review text: The authors establish necessary and sufficient conditions for the discreteness of the spectrum of the Laplacian acting on functions on a noncompact Riemannian manifold \(M\). In the article the Laplacian is defined as the self-adjoint operator associated to the bilinear form \[ a(u,v)=\int_M \langle du,dv\rangle \] defined on the Sobolev space \(W^{1,2}(M)\), viewed as a subspace in \(L^2(M)\). If \(M\) is complete, this is the Friedrichs extension of the classical Laplacian. The underlying manifolds are not assumed to be complete, so in particular any open subset of euclidean space is allowed as well. If \(M\) is the interior of a compact Riemannian manifold with boundary, the Laplacian thus obtained is the classical Laplace operator with Neumann boundary condition. The discreteness of the Laplacian is equivalent to the compactness of the Sobolev embedding \(W^{1,2}(M)\to L^2(M)\). The necessary and sufficient condition for the discreteness of the spectrum is formulated in terms of the isocapacitary function \(\nu_M\), defined in the article. Namely, in the case that \(M\) has infinite volume, one defines \(\nu_M(s)\) as the infimum of the capacity \(P(E)\) where the infimum runs over all subsets \(E\) of \(M\) of volume at least \(s\). In the case that \(M\) has finite volume, the definition is similar, see the article under review. The isocapacitary function is tightly related to the isoperimetric function \(s\mapsto \lambda_M(s)\), which is defined as the infimum of all perimeters \(P(E)\) of the subset \(E\) of \(M\) with \(s\leq \mathrm{vol}(E)\leq \mathrm{vol}(M)/2\). In the case of finite volume the authors show that the spectrum of the Laplacian is discrete if and only if \[ \lim_{s\to 0} \frac{s}{\nu_M(s)}= 0. \] In the infinite volume case it is discrete if and only if \[ \lim_{s\to 0} \frac{s}{\nu_M(s)}= 0\quad\mathrm{and}\quad\lim_{s\to \infty} \frac{s}{\nu_M(s)}= 0. \] As a corollary one sees that the spectrum is discrete in the finite volume case if \…
reviewed by: Bernd Ammann
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 5961186
zbMATH Keywords: discrete spectrum
zbMATH Keywords: Laplace operator
zbMATH Keywords: noncompact Riemannian manifold
MaRDI profile type: Publication
OpenAlex ID: W1484478872
Wikidata QID: Q115170788
DOI: 10.4310/JDG/1312998232
description: scientific article; zbMATH DE number 5961186