Convergence Of Alexandrov Spaces And Spectrum Of Laplacian

. Denote by A(n) the family of compact n-dimensional Alexandrov spaces with curvature \Gamma1, and k (M) the k th - eigenvalue of the Laplacian on M 2 A(n). We prove the continuity of k : A(n) ! R with respect to the Gromov-Hausdorff topology for each k; n 2 N. 1. Introduction For n 2 N and D ? 0, let M Ric (n; D) denote the family of isometry classes of closed n-dimensional Riemannian manifolds with Ricci curvature Ric M \Gamma(n \Gamma 1) and diameter diam(M) D, and M jsecj (n; D) the family of all M 2 M Ric (n; D) whose sectional curvatures KM satisfy jKM j 1. Fukaya [5] proved that, if we equip each M 2 M jsecj (n; D) with normalized volume measure dvol M = vol(M ), then for each k 2 N the function k which assigns to each M 2 M jsecj (n; D) the k th -eigenvalue of the Laplacian on M extends to a continuous function on the measured Gromov-Hausdorff closure of M jsecj (n; D). He also conjectured this statement to be true even if M jsecj (n; D) is replaced with M Ric ..