Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian
Mantuano, Tatiana
In: Annals of Global Analysis and Geometry, 2005, vol. 27, no. 1, p. 33-46
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- For κ ⩾ 0 and r0 > 0 let ℳ(n, κ, r0) be the set of all connected, compact n-dimensional Riemannian manifolds (Mn, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r0. We study the relation between the kth eigenvalue λk(M) of the Laplacian associated to (Mn,g), Δ = −div(grad), and the kth eigenvalue λk(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r0) such that for all M ∈ ℳ(n, κ, r0) and X a discretization of $${M, c \leqslant \frac{\lambda_{k}(M)}{\lambda_{k}(X)} \leqslant C}$$ for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r0) such that the Gromov-Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r0 such that $${c \leqslant \frac{\lambda_{k}(M)}{\lambda_{k}(N)} \leqslant C}$$ for all $${k \in \mathbb{N}}$$