Inverse spectral positivity for surfaces

Abstract : Let $(M,g)$ be a complete non-compact Riemannian surface. We consider operators of the form $\Delta + aK + W$, where $\Delta$ is the non-negative Laplacian, $K$ the Gaussian curvature, $W$ a locally integrable function, and $a$ a positive real number. Assuming that the positive part of $W$ is integrable, we address the question ''What conclusions on $(M,g)$ and $W$ can one draw from the fact that the operator $\Delta + aK + W$ is non-negative ?'' As a consequence of our main result, we get a new proof of Huber's theorem and Cohn-Vossen's inequality, and we improve earlier results in the particular cases in which $W$ is non-positive and $a = 1/4$ or $a \in (0,1/4)$.
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Revista Matemática Iberoamericana, European Mathematical Society, 2014, 30, pp.1237-1264. <http://www.ems-ph.org/journals/journal.php?jrn=RMI>. <10.4171/RMI/813>
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Pierre Bérard, Philippe Castillon. Inverse spectral positivity for surfaces. Revista Matemática Iberoamericana, European Mathematical Society, 2014, 30, pp.1237-1264. <http://www.ems-ph.org/journals/journal.php?jrn=RMI>. <10.4171/RMI/813>. <hal-00644783v4>

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