Connected essential spectrum: the case of differential forms

TL;DR

This paper proves that on certain complete manifolds with vanishing curvature at infinity, the essential spectrum of the Hodge Laplacian on differential forms is a connected interval, using Gromov-Hausdorff convergence and a generalized Weyl criterion.

Contribution

It introduces a new approach to analyze the essential spectrum of the Hodge Laplacian on differential forms on manifolds with specific asymptotic curvature conditions.

Findings

Essential spectrum is a connected interval for manifolds with vanishing curvature at infinity.

Under weaker Ricci curvature conditions, the essential spectrum on certain forms is [0,∞).

The method involves Gromov-Hausdorff convergence and a generalized Weyl criterion.

Abstract

In this article we prove that, over complete manifolds of dimension $n$ with vanishing curvature at infinity, the essential spectrum of the Hodge Laplacian on differential $k$-forms is a connected interval for $0\leq k\leq n$. The main idea is to show that large balls of these manifolds, which capture their spectrum, are close in the Gromov-Hausdorff sense to product manifolds. We achieve this by carefully describing the collapsed limits of these balls. Then, via a new generalized version of the classical Weyl criterion, we demonstrate that very rough test forms that we get from the $\varepsilon$-approximation maps can be used to show that the essential spectrum is a connected interval. We also prove that, under a weaker condition where the Ricci curvature is asymptotically nonnegative, the essential spectrum on $k$-forms is $[0,\infty)$, but only for $0\leq k\leq q$ and $n-q \leq k\leq n$ for some integer $ q\geq 1$ which depends the structure of the manifolds at infinity.