Commentarii Mathematici Helvetici

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Volume 80, Issue 4, 2005, pp. 725–754
DOI: 10.4171/CMH/32

Published online: 2005-12-31

Spectral convergence of manifold pairs

Karsten Fissmer[1] and Ursula Hamenstädt[2]

(1) Bonn, Germany
(2) Universität Bonn, Germany

Let $(M_i,A_i)_i$ be pairs consisting of a complete Riemannian manifold $M_i$ and a nonempty closed subset $A_i$. Assume that the sequence $(M_i,A_i)_i$ converges in the Lipschitz topology to the pair $(M,A)$. We show that there is a number $c\geq 0$ which is determined by spectral properties of the ends of $M_i-A_i$ and such that the intersections with $[0,c)$ of the spectra of $M_i$ converge to the intersection with $[0,c)$ of the spectrum of $M$. 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.

Keywords: Laplace operator, spectrum, Lipschitz convergence, spectral convergence, multiplicities

Fissmer Karsten, Hamenstädt Ursula: Spectral convergence of manifold pairs. Comment. Math. Helv. 80 (2005), 725-754. doi: 10.4171/CMH/32