On the discreteness of the spectrum of the Laplacian on noncompact Riemannian manifolds
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 \[ \lim_{s\to 0} \frac{s}{\lambda_M(s)}= 0, \] however this is not a necessary condition. Similarly a sufficient condition in the infinite volume case is \[ \lim_{s\to 0} \frac{s}{\lambda_M(s)}= 0\quad\mathrm{and}\quad\lim_{s\to \infty} \frac{s}{\lambda_M(s)}= 0. \] The article is well written and contains some helpful examples. These examples show in particular, that the criterion in terms of the isoperimetric function is not necessary. It thus seems that the isocapacitary function is more suitable to study the discreteness of the Laplacian than the isoperimetric function, although the latter function has been studied much more in the literature. Note that in many articles the term ``isoperimetric profile'' is used instead of ``isoperimetric function''.
scientific article; zbMATH DE number 5961186