On the spectrum of non-compact manifolds with finite volume
Let M be a complete non-compact manifold with finite volume, and \(\lambda_ 0^{ess}\) the lower bound of the essential spectrum of the Laplacian on M \((\lambda_ 0^{ess}=\infty\) if M has discrete spectrum). We prove the estimates \((1/4)h^{ess^ 2}\leq \lambda_ 0^{ess}\leq(1/4)\mu^ 2_ f\) where \(h^{ess}\) is the essential Cheeger's isoperimetric constant, and \[ \mu_ f=\overline{\lim}_{r\to \infty}(-1/r) \log(Vol(M)-Vol(Br)) \] measures how quickly the volume of a ball of radius r approaches the volume of M. Techniques are presented for estimating these constants, and examples are given.
scientific article; zbMATH DE number 3854076