Spectral convergence of manifold pairs

  • Karsten Fissmer

    Bonn, Germany
  • Ursula Hamenstädt

    Universität Bonn, Germany


Let (Mi,Ai)i(M_i,A_i)_i be pairs consisting of a complete Riemannian manifold MiM_i and a nonempty closed subset AiA_i. Assume that the sequence (Mi,Ai)i(M_i,A_i)_i converges in the Lipschitz topology to the pair (M,A)(M,A). We show that there is a number c0c\geq 0 which is determined by spectral properties of the ends of MiAiM_i-A_i and such that the intersections with [0,c)[0,c) of the spectra of MiM_i converge to the intersection with [0,c)[0,c) of the spectrum of MM. 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. 725–754

DOI 10.4171/CMH/32