An inverse spectral problem on surfaces

Abstract

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.