Abstract

We consider the Monge-Ampère-type equation $\det = \mathrm{const}.$, where $A$ is the Schouten tensor of a conformally related metric and $\lambda > 0$ is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique