On the spectral stability of finite coverings

TL;DR

This paper investigates the spectral stability of finite coverings of Riemannian manifolds, proving the absence of new eigenvalues in certain spectral intervals under specific geometric and algebraic conditions.

Contribution

It establishes conditions under which finite coverings do not introduce new eigenvalues below the essential spectrum, advancing understanding of spectral invariance in geometric analysis.

Findings

No new eigenvalues in [0,Λ] for certain finite coverings

Spectral stability depends on fundamental group representation properties

Results apply to both specific and random coverings

Abstract

We prove the non-existence of new eigenvalues in $[0,\Lambda]$ for specific and random finite coverings of a complete and connected Riemannian manifold $M$ with Ricci curvature bounded from below, where $\Lambda$ is any positive number below the essential spectrum of $M$ and the spectrum of the universal cover of $M$, provided the representation theory of the fundamental group of $M$ satisfies certain conditions.