Let be a complete noncompact Riemannian surface. We consider operators of the form , where is the nonnegative Laplacian, the Gaussian curvature, a locally integrable function, and a positive real number. Assuming that the positive part of is integrable, we address the question "What conclusions on and on can one draw from the fact that the operator 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 is nonpositive and or .
Cite this article
Pierre Bérard, Philippe Castillon, Inverse spectral positivity for surfaces. Rev. Mat. Iberoam. 30 (2014), no. 4, pp. 1237–1264DOI 10.4171/RMI/813