Convergence of Alexandrov spaces and spectrum of Laplacian. (English) Zbl 0974.58030

Let \(A(n)\) be the collection of isometry classes of compact Alexandrov spaces of dimension \(n\) with curvature bounded from below by \(-1\). Give \(A(n)\) the Gromov-Hausdorff topology [see K. Fukaya, Invent. Math. 87, 517-547 (1987; Zbl 0589.58034)]. A. Kasue and H. Kumura [Tôhoku Math. J. (2) 46, 147-179 (1994; Zbl 0814.53035) and ibid. 48, 71-120 (1996; Zbl 0853.58100)] have defined the spectral topology on \(A(n)\). Let \(\lambda_k\) be the \(k\)-th eigenvalue of the Laplacian on an element of \(A(n)\). The author shows that \(\lambda_k:A(n)\rightarrow R\) is a continuous function for each \((k,n)\) by showing the Gromov-Hausdorff topology coincides with the spectral topology on \(A(n)\).


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20 Global Riemannian geometry, including pinching
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