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Eigenvalue Estimates of the Hodge Laplacian Under Lower Ricci Curvature Bound

Anusha Bhattacharya, Soma Maity, Aditya Tiwari

TL;DR

The paper advances spectral geometry for differential forms by deriving uniform eigenvalue bounds for the Hodge Laplacian under a Ricci curvature lower bound, positive injectivity radius, and bounded diameter. It extends Cheng-type and discretization methods—originally developed under sectional curvature bounds—to the weaker Ricci-bound setting via a Čech–de Rham framework, harmonic-coordinate technology, and a local spectral gap. The authors establish both lower and upper bounds for the eigenvalues $\lambda_{k,p}$, with quantitative dependence on dimension, Ricci bound $\xi$, injectivity radius $r_0$, and diameter $D$, and they apply these results to obtain bounds for the connection Laplacian on $1$-forms, a global Poincaré inequality for forms, and a spectral equivalence between smooth and discrete spectra. The approach hinges on discretizing the manifold with controlled covers, constructing a discretization operator $\mathcal{D}$, and leveraging local-to-global transfer through domain decomposition and volume comparison. Overall, the work broadens the scope of eigenvalue estimates in geometric analysis and provides practical tools for comparing smooth and discrete spectra under Ricci-controlled geometries.

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 Poincaré inequality for differential forms under the same geometric assumptions.

Eigenvalue Estimates of the Hodge Laplacian Under Lower Ricci Curvature Bound

TL;DR

The paper advances spectral geometry for differential forms by deriving uniform eigenvalue bounds for the Hodge Laplacian under a Ricci curvature lower bound, positive injectivity radius, and bounded diameter. It extends Cheng-type and discretization methods—originally developed under sectional curvature bounds—to the weaker Ricci-bound setting via a Čech–de Rham framework, harmonic-coordinate technology, and a local spectral gap. The authors establish both lower and upper bounds for the eigenvalues , with quantitative dependence on dimension, Ricci bound , injectivity radius , and diameter , and they apply these results to obtain bounds for the connection Laplacian on -forms, a global Poincaré inequality for forms, and a spectral equivalence between smooth and discrete spectra. The approach hinges on discretizing the manifold with controlled covers, constructing a discretization operator , and leveraging local-to-global transfer through domain decomposition and volume comparison. Overall, the work broadens the scope of eigenvalue estimates in geometric analysis and provides practical tools for comparing smooth and discrete spectra under Ricci-controlled geometries.

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 -forms and establish a global Poincaré inequality for differential forms under the same geometric assumptions.
Paper Structure (11 sections, 23 theorems, 136 equations, 1 figure)

This paper contains 11 sections, 23 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Harmonic Coordinates
  4. Discretization
  5. Čech Cochains and Norms
  6. Lower Bound of the Hodge Laplacian Eigenvalues
  7. Local absolute spectral gap under Ricci Bounds
  8. The Discretization Operator
  9. Upper Bound of the Hodge Laplacian Eigenvalues
  10. Proof of Theorem \ref{['thm:main_theorem']}:
  11. Some Applications

Key Result

Theorem 1.1

Let $(M,g)$ be a closed Riemannian $n$-manifold with ${\rm Ric} \geq (n-1)\xi$, ${\rm inj}(M)\geq r_0$, ${\rm diam}(M) \leq D<\infty$. Then there exist positive constants $c(n,D,\xi,r_0,p)$ and $c'(n,D,\xi,r_0,p)$ such that where $V_{\xi}(D)$ denotes the volume of a ball of radius $D$ in a space of constant curvature $\xi.$

Figures (1)

  • Figure 1: Čech-de Rham double complex and the inductive construction of $\mathcal{D}(\omega)$.

Theorems & Definitions (48)

  • Conjecture : Honda-Mondino
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Definition 2.1: $\varepsilon$-discretization
  • Definition 2.2
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 38 more