Discretization of Riemannian manifolds applied to the Hodge Laplacian

2006/09/21 by Mantuano, Tatiana · 2 citations
#53C20 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.math/0609599

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.

Cited by

• Coexact 1-Laplacian spectral gap and exponential growth of a group
• Small eigenvalues of the Hodge-Laplacian with sectional curvature bounded below

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