Hokkaido Mathematical Journal Vol. 20 (1991) p. 265-278 Prescribing Ricci curvature on open surfaces Jianguo CAO1 and Dennis DETURCK2 Dedicated to Professor Nobom Tanaka on the occasion of his sixtieth birthday (Received May 8, 1990, Revised August 6, 1990) 1. Introduction. This work grew out of an attempt to understand the problem of finding a complete, rotationally symmetric metric on R^{n} for n\geq 3 with a prescribed, rotationally symmetric Ricci tensor. As will be reported else- where, one runs headlong into the problem of deciding whether a given “Ricci candidate” on a surface (in particular, on R^{2} ) is in fact the Ricci tensor of a complete metric on the surface. The issue of completeness is surprisingly delicate, even for surfaces, and so the subject of this paper is the following: PROBLEM. Given a symmetric covariant tensor R of rank two on an open surface S (of fifinite topological type), when does there exist a complete metric g such that Ric(g)=R on S ? Of course, by Ric(\#) , we mean the (covariant) Ricci tensor of the metric g . Once one has existence of such a metric, it becomes reasonable to ask to what extent the metric is unique (since Ric(cg)=Ric(g) for any positive constant c , the best uniqueness statement possible is “unique up to a con- fact multipl\"e). As we shall see below (Lemma 1. 4), it can be directly determined from the tensor R whether the metric we seek will have finite or infinite total curvature. We give a fairly complete solution of the prob- lem in the finite-total-curvature case (Theorem 2. 5), including uniqueness statements, and we give sufficient conditions for the existence of a com- plete g in the infinite-total-curvature case (Theorem 3. 3). Examples 3. 1 and 3. 2 indicate why the issue of completeness in the latter case is so delicate. On tw0-dimensional manifolds, one can take advantage of the fact that the Ricci tensor is equal to the Gauss curvature times the metric ten- 1 Supported by NSF Grant DMS-8610730 at the Insititute for Advanced Study 2 Supported by NSF Grant DMS-9001707 and the Institute for Advanced Study