Inverse spectral positivity for surfaces

  • Pierre Bérard

    Université Grenoble I, Saint-Martin-d'Hères, France
  • Philippe Castillon

    Université de Montpellier II, France


Let (M,g)(M,g) be a complete noncompact Riemannian surface. We consider operators of the form Δ+aK+W\Delta + aK + W, where Δ\Delta is the nonnegative Laplacian, KK the Gaussian curvature, WW a locally integrable function, and aa a positive real number. Assuming that the positive part of WW is integrable, we address the question "What conclusions on (M,g)(M,g) and on WW can one draw from the fact that the operator Δ+aK+W\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 WW is nonpositive and a=1/4a = 1/4 or a(0,1/4)a \in (0,1/4).

Cite this article

Pierre Bérard, Philippe Castillon, Inverse spectral positivity for surfaces. Rev. Mat. Iberoam. 30 (2014), no. 4, pp. 1237–1264

DOI 10.4171/RMI/813