Inverse spectral positivity for surfaces

Abstract

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 $a\in (0,1/4)$.