Normal covering spaces with maximal bottom of spectrum

TL;DR

This paper investigates spectral-tightness in Riemannian manifolds, showing it is a topological property linked to the fundamental group and characterizing when non-positively curved manifolds are spectrally-tight based on their Euclidean factors.

Contribution

It establishes spectral-tightness as a topological property and characterizes it for non-positively curved manifolds, extending existing results on the spectrum of Riemannian coverings.

Findings

Spectral-tightness is a topological property determined by the fundamental group.

Non-positively curved, closed manifolds are spectrally-tight iff their Euclidean local de Rham factor has zero dimension.

Results extend the understanding of the bottom of the spectrum in Riemannian coverings.

Abstract

We study the property of spectral-tightness of Riemannian manifolds, which means that the bottom of the spectrum of the Laplacian separates the universal covering space from any other normal covering space of a Riemannian manifold. We prove that spectral-tightness of a closed Riemannian manifold is a topological property characterized by its fundamental group. As an application, we show that a non-positively curved, closed Riemannian manifold is spectrally-tight if and only if the dimension of its Euclidean local de Rham factor is zero. In their general form, our results extend the state of the art results on the bottom of the spectrum under Riemannian coverings.