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 where is the Laplacian of the surface, is its curvature and 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.
Cite this article
Philippe Castillon, An inverse spectral problem on surfaces. Comment. Math. Helv. 81 (2006), no. 2, pp. 271–286DOI 10.4171/CMH/52