Eigenvalue Estimates of the Hodge Laplacian Under Lower Ricci Curvature Bound

TL;DR

This paper derives uniform bounds for the eigenvalues of the Hodge Laplacian on differential forms on closed Riemannian manifolds with lower Ricci curvature bounds, extending previous results to more general curvature conditions.

Contribution

It extends eigenvalue estimates of the Hodge Laplacian from bounded sectional curvature to manifolds with only lower Ricci curvature bounds, including applications to connection Laplacian and Poincaré inequalities.

Findings

Uniform eigenvalue bounds for the Hodge Laplacian

Eigenvalue bounds for the connection Laplacian on 1-forms

Global Poincaré inequality for differential forms

Abstract

We establish uniform lower and upper bounds for the eigenvalues of the Hodge Laplacian acting on differential forms on closed Riemannian manifolds with a lower Ricci curvature bound, a positive lower bound on the injectivity radius, and an upper bound on the diameter. Our results extend earlier work of Dodziuk, Lott, and Mantuano, which required bounded sectional curvature, to the broader setting of lower Ricci curvature bounds. As applications, we obtain uniform eigenvalue bounds for the connection Laplacian acting on $1$-forms and establish a global Poincar\'e inequality for differential forms under the same geometric assumptions.