Inverse spectral positivity for surfaces

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).