A Cheng-type Eigenvalue-Comparison Theorem for the Hodge Laplacian

TL;DR

This paper establishes a uniform upper bound for the eigenvalues of the Hodge Laplacian on certain closed Riemannian manifolds, extending classical eigenvalue comparison results to differential forms under Ricci curvature and injectivity radius bounds.

Contribution

It extends Cheng's eigenvalue comparison theorem to the Hodge Laplacian on differential forms, relaxing curvature conditions to Ricci bounds and injectivity radius constraints.

Findings

Derived a uniform upper bound for Hodge Laplacian eigenvalues.

Extended eigenvalue comparison to differential forms under Ricci bounds.

Provided applications to connection Laplacian eigenvalues on 1-forms.

Abstract

We consider the class of closed Riemannian $n$-manifolds with Ricci curvature and injectivity radius bounded below by uniform constants, and an upper bound on the diameter. We establish a uniform upper bound for the eigenvalues of the Hodge Laplacian acting on differential forms on Riemannian manifolds in this class, similar to the classical eigenvalue comparison theorem proved by Cheng for the Laplace-Beltrami operator acting on smooth functions. This extends earlier work of Dodziuk and Lott, which required sectional curvature bounds in addition to bounds on other geometric quantities. As an application, we obtain uniform eigenvalue estimates for the connection Laplacian acting on $1$-forms.