Prescribing curvature on open surfaces (Q1204226)

This paper is an expanded version of a previous announcement [C.R. Acad. Sci., Paris, Sér. I, 310, No. 4, 203-206 (1990; Zbl 0697.53039)] and deals with the problem of prescribing the curvature on a Riemann surface, that is on an oriented surface equipped with a conformal class of Riemannian metrics. The authors are interested in non compact surfaces, since in the compact case, the theory is well understood [\textit{J. L. Kazdan} and \textit{F. W. Warner}, Ann. Math., II. Ser. 99, 14-47 (1974; Zbl 0273.53034)]. They prove that on any connected non compact Riemann surface of finite type, different from \({\mathbb{C}}\) or \({\mathbb{C}}^*\), there is no obstruction to construct a conformal metric with prescribed curvature. However, the metric is usually not complete. Even if one restricts to complete metrics, the problem of prescribing the curvature may have a continuum of solutions, with variable asymptotic geometries (see the example in their above mentioned paper). Thus, they are led to study complete metrics, keeping control on their asymptotic geometries. The authors propose two precise formulations for the problem of prescribing the curvature on open Riemann surfaces: Problem 1 is the natural one for surfaces with finite total curvature, while Problem 2 deals with surfaces having complete hyperbolic ends. Thanks to the work of A. Huber, the authors' answer to Problem 1 is expressed in a form very similar to the classical results available for compact surfaces. When the prescribed curvature function is negatively pinched at infinity, under suitable assumptions, the authors are able to solve Problem 2. The technique used to investigate Problems 1 and 2 is by now classical.