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Discretization of Riemannian manifolds applied to the Hodge Laplacian

Tatiana Mantuano
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Abstract

An adapted version of the proof (due to A. Weil) of the well-known de Rham Theorem allows us to compare uniformly the spectrum of the Hodge Laplacian acting on differential forms (on a compact Riemannian manifold) to the spectrum of the combinatorial Laplacian acting on cochains associated to an open cover (made of balls of sufficiently small radius). We exhibit then a lower bound for the first positive eigenvalue of the combinatorial Laplacian and deduce a lower bound for the first positive eigenvalue of the Hodge Laplacian.

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Tatiana Mantuano. 2006-09-21. Discretization of Riemannian manifolds applied to the Hodge Laplacian. https://arxiv.org/abs/math/0609599

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