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

Shioya, Takashi

J. Math. Soc. Japan 53, No. 1, 1-15 (2001).

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 \(A(n)\).

Reviewer: Peter B.Gilkey (Eugene)

MSC:

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

Keywords:

Alexandrov space; eigenvalue; spectrum; Laplacian; Gromov-Hausdorff distance; spectral distance

Citations:

Zbl 0589.58034; Zbl 0814.53035; Zbl 0853.58100