Spectral positivity and Riemannian coverings

Abstract

Let $(M,g)$ be a complete non-compact Riemannian manifold. We consider operators of the form ${\Delta }_g+V$, where ${\Delta }_g$ is the non-negative Laplacian associated with the metric $g$, and $V$ a locally integrable function. Let $\rho :(\hat{M},\hat{g})\to (M,g)$ be a Riemannian covering, with Laplacian ${\Delta }_{\hat{g}}$ and potential $\hat{V}=V\circ \rho$. If the operator $\Delta +V$ is non-negative on $(M,g)$, then the operator ${\Delta }_{\hat{g}}+\hat{V}$ is non-negative on $(\hat{M},\hat{g})$. In this note, we show that the converse statement is true provided that ${\pi }_1(\hat{M})$ is a co-amenable subgroup of ${\pi }_1(M)$.

Final version to appear in the Bulletin of the London Mathematical Society