Prescribing curvature problem of Bakry-Émery Ricci tensor

We consider the problem of deforming a metric in its conformal class on a closed manifold, such that the k-curvature defined by the Bakry-mery Ricci tensor is a constant. We show its solvability on the manifold, provided that the initial Bakry-mery Ricci tensor belongs to a negative cone. Moveover, the Monge-Ampère type equation with respect to the Bakry-mery Ricci tensor is also considered.