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Article Dans Une Revue Revista Matemática Iberoamericana Année : 2014

Inverse spectral positivity for surfaces

Résumé

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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Dates et versions

hal-00644783 , version 1 (25-11-2011)
hal-00644783 , version 2 (23-03-2012)
hal-00644783 , version 3 (30-06-2012)
hal-00644783 , version 4 (23-03-2015)

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Citer

Pierre Bérard, Philippe Castillon. Inverse spectral positivity for surfaces. Revista Matemática Iberoamericana, 2014, 30 (4), pp.1237-1264. ⟨10.4171/rmi/813⟩. ⟨hal-00644783v4⟩
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