Prescribing Ricci curvature on open surfaces
Suppose a tensor \(R\) on an open surface of finite topological type satisfies the (simple) local conditions to be the Ricci tensor of a Riemannian metric. This paper asks if there exists a globally defined complete metric \(g\) which satisfies these conditions. The tensor \(R\) determines, whether \(g\) will have finite or infinite total curvature. In the former case, a criterion is given, including uniqueness statements, using the work of \textit{A. Huber} [Comment. Math. Helv. 32, 13--72 (1957; Zbl 0080.15001), ibid. 41, 105--136 (1966; Zbl 0161.09004)] and \textit{R. Finn} [ibid. 40, 1--30 (1965; Zbl 0192.27301)]. The paper contains useful examples, including those illustrating the delicacy of determining completeness in the case of infinite total curvature. In this case, sufficient conditions are given.
scientific article; zbMATH DE number 11094
scientific article