Discretization of Riemannian manifolds applied to the Hodge Laplacian

Details Discretization of Riemannian manifolds applied to the Hodge Laplacian Discretization of Riemannian manifolds applied to the Hodge Laplacian

2006-09-21

arxiv.org/abs/math/0609599v1

Mathematics

Differential Geometry

Scientific paper

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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