# Commentarii Mathematici Helvetici

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**Volume 81, Issue 2, 2006, pp. 271–286**

**DOI: 10.4171/CMH/52**

An inverse spectral problem on surfaces

Philippe Castillon^{[1]}(1) Dept des Sc. Mathématiques, Université de Montpellier II, 34095, Montpellier CEDEX 5, France

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.

*Keywords: *Spectral theory, minimal surfaces, stability operator

Castillon Philippe: An inverse spectral problem on surfaces. *Comment. Math. Helv.* 81 (2006), 271-286. doi: 10.4171/CMH/52