JournalscmhVol. 81 , No. 2DOI 10.4171/cmh/52

An inverse spectral problem on surfaces

  • Philippe Castillon

    Université de Montpellier II, France
An inverse spectral problem on surfaces cover

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 Δ+λK\Delta+\lambda\mathcal{K} where Δ\Delta is the Laplacian of the surface, K\mathcal{K} is its curvature and λ\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.