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Pré-Publication, Document De Travail Année : 2012

Spectral positivity and Riemannian coverings

Résumé

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 $\rho : (\widehat{M},\hat{g}) \to (M,g)$ be a Riemannian covering, with Laplacian $\Delta_{\hat{g}}$ and potential $\widehat{V} = V \circ \rho$. If the operator $\Delta + V$ is non-negative on $(M,g)$, then the operator $\Delta_{\hat{g}} + \widehat{V}$ is non-negative on $(\widehat{M},\hat{g})$. In this note, we show that the converse statement is true provided that $\pi_1(\widehat{M})$ is a co-amenable subgroup of $\pi_1(M)$.
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Dates et versions

hal-00682177 , version 1 (23-03-2012)
hal-00682177 , version 2 (30-06-2012)
hal-00682177 , version 3 (04-03-2013)

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Pierre Bérard, Philippe Castillon. Spectral positivity and Riemannian coverings. 2012. ⟨hal-00682177v2⟩
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