The purpose of this paper is to prove how the positivity of some operators on a Riemannian surface gives informations on the conformal type of the surface (the operators considered here are of the form \Delta+\lambda\mathcal{K} where \Delta is the Laplacian of the surface, \mathcal{K} is its curvature and \lambda is a real number). In particular we obtain a theorem “à la Huber”: under a spectral hypothesis we prove that the surface is conformally equivalent to a Riemann surface with a finite number of points removed. This problem has its origin in the study of stable minimal surfaces.