Discretization of compact Riemannian manifolds applied to the spectrum of Laplacian

For kappa >= 0 and r(0) > 0 let M(n, kappa, r(0)) be the set of all connected, compact n-dimensional Riemannian manifolds (M-n, g) with Ricci (M, g) >= -(n - 1)kappa g and Inj (M) >= r(0). We study the relation between the kth eigenvalue lambda(k) ( M) of the Laplacian associated to (Mn, g), Delta = -div(grad), and the kth eigenvalue lambda(k) (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,. and r0) such that for all M is an element of M(n, kappa, r(0)) and X a discretization of M, c <= lambda(k)(M)/lambda(k)(X) <= C for all k < vertical bar X vertical bar. Then, we obtain the same kind of result for two compact manifolds M and N is an element of M(n, kappa, r(0)) such that the Gromov-Hausdorff distance between M and N is smaller than some eta > 0. We show that there exist constants c, C > 0 depending on eta, n, kappa and r(0) such that c <= lambda(k)(M)/lambda(k)(N) <= C for all k is an element of N.