Complete manifolds with bounded curvature and spectral gaps

TL;DR

This paper investigates the spectral properties of complete noncompact manifolds with bounded curvature, demonstrating conditions under which their essential spectrum can have arbitrarily many gaps, including constructions for specific metrics.

Contribution

It provides new criteria and constructions for manifolds to have arbitrarily large finite numbers of spectral gaps in their essential spectrum.

Findings

Existence of metrics with arbitrarily many spectral gaps

Conditions for essential spectrum gaps on noncompact manifolds

Construction of equivalent metrics with spectral gaps

Abstract

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the base so that the lifted metric has an arbitrarily large finite number of gaps in its essential spectrum. Also, for any complete noncompact manifold with bounded curvature and positive injectivity radius we construct a metric uniformly equivalent to the given one (also of bounded curvature and positive injectivity radius) with an arbitrarily large finite number of gaps in its essential spectrum.