Eigenvalue Estimates of the Hodge Laplacian Under Lower Ricci Curvature Bound

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.