Inverse spectral positivity for surfaces
Résumé
Let $(M,g)$ be a complete non-compact Riemannian surface. We consider operators of the form $\Delta + aK - q$, where $\Delta$ is the non-negative Laplacian, $K$ the Gaussian curvature, $q$ a non-negative function, and $a$ a positive real number. We address the question ''What conclusions on $(M,g)$ and $q$ can one draw from the fact that the operator $\Delta + aK - q$ is non-negative'' and we improve earlier results in particular in the cases $a = \4$ and $a \in (0,\4)$. We also show that the non-negativity is preserved under normal Riemannian covering with amenable covering group.
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