Convergence of Alexandrov spaces and spectrum of Laplacian

Abstract

Denote by $\mathcal{A}\left(n\right)$ the family of isometry classes of compact n-dimensional Alexandrov spaces with curvature $\ge -1$, and ${\lambda }_k\left(M\right)$ the $k^{th}$ eigenvalue of the Laplacian on $M\in \mathcal{A}\left(n\right)$. We prove the continuity of ${\lambda }_k:\mathcal{A}\left(n\right)$$\to R$ with respect to the GromovHausdorff topology for each $k,$$n\in N$, and moreover that the spectral topology introduced by Kasue-Kumura [7], [8] coincides with the Gromov-Hausdorff topology on $\mathcal{A}\left(n\right)$.