Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Anusha Bhattacharya, Soma Maity

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.

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

Abstract

We consider the class of closed Riemannian -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 -forms.
Paper Structure (6 sections, 14 theorems, 55 equations)

This paper contains 6 sections, 14 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Harmonic Coordinates
  4. Upper Bound of the Hodge Laplacian Eigenvalues
  5. Proof of Theorem \ref{['thm:main_theorem']}:
  6. Bound on spectral gap for non-compact manifolds

Key Result

Theorem 1.1

Cheng Suppose $M$ is a closed Riemannian manifold with dimension $n$ and diameter $D$. If Ricci curvature of $M$ is greater than or equal to $(n-1)\xi$ then where $\lambda_0^D(B_{\xi}\left(r)\right)$ denotes the first positive Dirichlet eigenvalue of the Laplacian on a ball of radius $r$ in the space of constant curvature $\xi$ and dimension $n$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Definition 2.1: $\varepsilon$-discretization
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Lemma 3.1
  • proof
  • ...and 16 more