SCHR ¨ ODINGER OPERATORS ASSOCIATED TO A HOLOMORPHIC MAP Sebasti´ an Montiel* Antonio Ros* Universidad de Granada Spain In this work we will expose certain ideas and results concerning a kind of Schr¨ odinger operators which can be considered on a compact Riemann surface. These operators will be constructed by using as a potential the energy density of a holomorphic map from the surface to the two-sphere. Besides the interest that their study has from an analytical point of view, we will see that they appear, in a natural way, in different geometrical situations such as the study of the index of complete minimal surfaces with finite total curvature and the study of the critical points of the Willmore functional. This paper is, in fact, an expanded version of an invited lecture given by the first author in the Global Differential Geometry and Global Analysis Conference held at the Technische Universit¨at of Berlin in June, 1990. Introduction and preliminaries Let Σ be a compact Riemann surface and φ :Σ → S 2 a holomorphic map from this surface to the unit two–sphere S 2 . Consider any metric ds 2 on Σ compatible with the complex structure and let ∆ and ∇ be its Laplacian and gradient respectively. Having chosen this metric, one has the following Schr¨ odinger operator (1-1) L =∆+ |∇φ| 2 . Our aim here is to study spectral properties of these operators and relate them to the map φ and the surface Σ. Of course, the eigenvalues and eigenfunctions of such an operator L depend strongly on the metric ds 2 . However, we want to obtain information from it which only refers to φ and Σ. This can be done in two ways: first, by looking for spectral properties of L which are independent on the chosen metric; or, second, by putting on Σ a particular metric especially related to our problem. First, denote by Q φ the quadratic form corresponding to the self–adjoint operator L, that is (1-2) Q φ (u, u)=  Σ  |∇u| 2 - |∇φ| 2 u 2  dA u ∈ W 1 (Σ) *Partially supported by a DGCICYT Grant No. PS87–0115 Typeset by A M S-T E X 1