Let be pairs consisting of a complete Riemannian manifold and a nonempty closed subset . Assume that the sequence converges in the Lipschitz topology to the pair . We show that there is a number which is determined by spectral properties of the ends of and such that the intersections with of the spectra of converge to the intersection with of the spectrum of . This is used to construct manifolds with nontrivial essential spectrum and arbitrarily high multiplicities for an arbitrarily large number of eigenvalues below the essential spectrum.
Cite this article
Karsten Fissmer, Ursula Hamenstädt, Spectral convergence of manifold pairs. Comment. Math. Helv. 80 (2005), no. 4, pp. 725–754DOI 10.4171/CMH/32