Abstract

CASTILLON, PHILIPPE. Spectral properties and conformal type of surfaces. An. Acad. Bras. Ciênc. [online]. 2002, vol.74, n.4, pp. 585-588. ISSN 0001-3765.  http://dx.doi.org/10.1590/S0001-37652002000400003.

In this short note, we announce a result relating the geometry of a riemannian surface to the positivity of some operators on this surface (the operators considered here are of the form surface Laplacian plus a scalar multiple of the curvature function). 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.