Spectral convergence of manifold pairs

• Autores: Karsten Fissmer, Ursula Hamenstädt
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 80, Nº 4, 2005, págs. 725-754
• Idioma: inglés
• DOI: 10.4171/cmh/32
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• Resumen

  • 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.