Revista Matemática Iberoamericana

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Volume 30, Issue 4, 2014, pp. 1237–1264
DOI: 10.4171/RMI/813

Published online: 2014-12-15

Inverse spectral positivity for surfaces

Pierre Bérard[1] and Philippe Castillon[2]

(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. Iberoam. 30 (2014), 1237-1264. doi: 10.4171/RMI/813