Discretization of Riemannian manifolds applied to the Hodge Laplacian

Abstract

For $\kappa \geq 0$ and $r_0 > 0$, let ${\Bbb M}(n,\kappa,r_0)$ be the set of all connected compact $n$-dimensional Riemannian manifolds such that $|K_g| \leq \kappa$ and $Inj(M,g) \geq r_0$. We study the relation between the $k^{{\rm th}}$ positive eigenvalue of the Hodge Laplacian on differential forms and the $k^{{\rm th}}$ positive eigenvalue of the combinatorial Laplacian associated to an open cover (acting on \v{C}ech cochains). We show that for a fixed sufficiently small $\varepsilon > 0$ there exist positive constants $c_1$ and $c_2$ depending only on $n$, $\kappa$ and $\varepsilon$ such that for any $M \in {\Bbb M}(n,\kappa,r_0)$ and for any $\varepsilon$-discretization $X$ of $M$ we have $c_1 \lambda_{k,p}(X) \leq \lambda_{k,p}(M) \leq c_2 \lambda_{k,p}(X)$ for any $k \leq K$ ($K$ depends on $X$). Moreover, we find a lower bound for the spectrum of the combinatorial Laplacian and a lower bound for the spectrum of the Hodge Laplacian.