Convergence of Alexandrov spaces and spectrum of Laplacian

Abstract

Denote by \mathscr{A}(n) the family of isometry classes of compact n-dimensional Alexandrov spaces with curvature ≥q-1, and λ_k(M) the k^th eigenvalue of the Laplacian on M∈ \mathscr{A}(n). We prove the continuity of λ_k.\ \mathscr{A}(n)→ \bm{R} with respect to the Gromov-Hausdorff topology for each k, n∈ \bm{N}, and moreover that the spectral topology in-troduced by Kasue-Kumura [{7}], [{8}] coincides with the Gromov-Hausdorff topology on \mathscr{A}(n).