Uniform lower bound for the first Hodge 1-form eigenvalue under lower Ricci, diameter, and volume bounds in any dimension

Prove that for every closed Riemannian n-manifold M with Ricci curvature bounded below by −1, diameter at most D, and volume at least v, the first positive eigenvalue of the Hodge Laplacian acting on 1-forms on M is bounded below by a positive constant depending only on n, D, and v; equivalently, show that there exists C(n, D, v) > 0 such that ν_H,1(M) ≥ C(n, D, v).

Background

The paper establishes a quantitative global Poincaré inequality for 1-forms on closed Riemannian four-manifolds under two-sided Ricci curvature bounds (|Ric| ≤ 1), an upper bound on diameter, and a positive lower bound on volume, yielding a positive lower bound on the first positive eigenvalue of the Hodge Laplacian on 1-forms.

Classical lower bounds for eigenvalues on p-forms often require stronger curvature assumptions (e.g., positive curvature operator or two-sided sectional curvature bounds). The conjecture aims to extend such a uniform spectral gap to all dimensions under only a lower Ricci curvature bound (Ric ≥ −1) together with diameter and volume controls. The authors note that a lower bound on volume cannot be removed in contrast to the function case, referencing [CC90].