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

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.

💡 Research Summary

The paper extends Cheng’s classical eigenvalue comparison theorem—originally formulated for the Laplace‑Beltrami operator on functions—to the Hodge Laplacian acting on differential p‑forms, under considerably weaker geometric hypotheses. The authors consider the class M(n, ξ, r₀) of closed n‑dimensional Riemannian manifolds whose Ricci curvature satisfies Ric ≥ (n‑1)ξ, whose injectivity radius is bounded below by r₀, and whose diameter is bounded above by a uniform constant D. No sectional curvature bounds are required.

The main result (Theorem 1.2) states that for any such manifold (M,g) and any integer k≥1,

• if k ≥ D²/r_H (where r_H is a uniform harmonic radius depending only on (n, ξ, r₀)), then
  λ_{k,p}(M) ≤ 2^{2p+1} λ_{D,0}(B_ξ(D²/k)),
• if k ≤ D²/r_H, then
  λ_{k,p}(M) ≤ 2^{2p+1} λ_{D,0}(B_ξ(r_H)).

Here λ_{k,p}(M) denotes the k‑th eigenvalue of the Hodge Laplacian on p‑forms, and λ_{D,0}(B_ξ(r)) is the first Dirichlet eigenvalue of the scalar Laplacian on a ball of radius r in the simply‑connected space form of constant curvature ξ. The factor 2^{2p+1} reflects the passage from functions to p‑forms.

The proof relies on three technical pillars:

1. Harmonic coordinates and uniform harmonic radius. Anderson’s theorem guarantees that any manifold in M(n, ξ, r₀) admits harmonic coordinate charts on balls of radius r_H with metric components satisfying ½ δ_{ij} ≤ g_{ij} ≤ 2 δ_{ij} and controlled L^q‑norm of their first derivatives. This provides a quasi‑Euclidean setting on each small ball.

2. ε‑discretization and domain decomposition. For a chosen ε < r_H, a finite set X⊂M is selected such that the balls B(x,2ε) cover M and are pairwise disjoint at the level of radius ε. Lemma 2.3 (domain decomposition) shows that the global eigenvalues can be bounded by the maximum of Dirichlet eigenvalues on the individual balls.

3. Local Dirichlet estimate for p‑forms (Lemma 3.1). Using the harmonic coordinates, the authors construct a test p‑form ω = f ω₀ where ω₀ = dy₁∧…∧dy_p is parallel (d*ω₀ = 0). The Rayleigh quotient for ω reduces to that of the scalar function f, up to the factor 2^{2p+1}. Consequently, the first Dirichlet eigenvalue for p‑forms on a ball B is bounded by 2^{2p+1} times the scalar Dirichlet eigenvalue on the corresponding model ball B_ξ(ε).

By placing the ε‑balls along a maximal geodesic of length D and choosing ε = D/(2k), the authors obtain the global bound stated in Theorem 1.2. The result recovers Cheng’s original bound when p = 0 and provides a uniform estimate for all p.

Several corollaries illustrate the theorem’s flexibility:

• Corollary 3.2 gives explicit formulas for non‑negative Ricci curvature, showing λ_{k,p} ≤ 2^{2p+1} n²π² k²/D² for large k and λ_{k,p} ≤ 2^{2p‑1} n²π²/r_H² for small k.
• Corollary 3.3 treats the case Ric ≥ ‑(n‑1)ξ, producing dimension‑dependent constants that involve ξ and the harmonic radius.
• Theorem 3.4 derives a volume‑dependent bound λ_{k,p} ≤ C(n,p)·(k+1)·(V/α_n)^{2/n}, where V is the total volume and α_n the volume of the unit sphere.
• Corollary 3.5 applies the Bochner formula to obtain an upper bound for the first non‑zero eigenvalue of the connection Laplacian on 1‑forms: λ_{C,1,1} ≤ 2^{2p+1} n²π²/(4 r_H²) under Ric ≥ 0.

The paper also addresses non‑compact manifolds. Defining σ_p(M) as the supremum of the L²‑spectrum of the Hodge Laplacian, the authors show (Corollary 4.1) that as the harmonic radius tends to infinity, σ_p(M) ≤ 2^{2p‑1}(n‑1)²ξ. Using an exhaustion by compact balls and Lemma 4.2, they prove (Theorem 4.3) that for any complete manifold with Ric ≥ (n‑1)ξ, there exists a uniform r_H such that σ_p(M) ≤ 2^{2p+1} λ_{D,0}(B_ξ(r_H)). This provides a Ricci‑only spectral gap estimate, extending earlier results that required sectional curvature bounds.

Overall, the paper introduces a clean, geometric‑analytic framework that replaces sectional curvature assumptions with a lower Ricci bound and a uniform injectivity radius. By exploiting harmonic coordinates, discretization, and a careful comparison of quadratic forms, the authors obtain Cheng‑type eigenvalue bounds for the Hodge Laplacian, explicit volume‑dependent estimates, and new spectral gap results for both compact and non‑compact manifolds. These contributions deepen the understanding of how coarse geometric data control the full spectrum of differential forms, and open avenues for further applications in geometric analysis, heat kernel estimates, and the study of manifolds with bounded Ricci curvature.