Revista Matemática Iberoamericana
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Published online: 2014-12-15
Inverse spectral positivity for surfacesPierre Bérard and Philippe Castillon (1) Université Grenoble I, Saint-Martin-d'Hères, France
(2) Université de Montpellier II, France
Let $(M,g)$ be a complete noncompact Riemannian surface. We consider operators of the form $\Delta + aK + W$, where $\Delta$ is the nonnegative 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 on $W$ can one draw from the fact that the operator $\Delta + aK + W$ is nonnegative?" As a consequence of our main result, we get new proofs of Huber's theorem and Cohn–Vossen's inequality, and we improve earlier results in the particular cases in which $W$ is nonpositive and $a = 1/4$ or $a \in (0,1/4)$.
Keywords: Spectral theory, positivity, minimal surface, constant mean curvature surface
Bérard Pierre, Castillon Philippe: Inverse spectral positivity for surfaces. Rev. Mat. Iberoamericana 30 (2014), 1237-1264. doi: 10.4171/RMI/813