Inverse spectral positivity for surfaces

• Autores: Pierre Bérard, Philippe Castillon
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 30, Nº 4, 2014, págs. 1237-1264
• Idioma: inglés
• DOI: 10.4171/RMI/813
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• Resumen

  • Let (M,g) be a complete noncompact Riemannian surface. We consider operators of the form Δ+aK+W, where Δ 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 Δ+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∈(0,1/4).