  In this paper, we study the problem of conformally deforming a metric on a
$3$-dimensional manifold $M^3$ such that its $k$-curvature equals to a
prescribed function, where the $k$-curvature is defined by the $k$-th
elementary symmetric function of the eigenvalues of the Einstein tensor, $1\le
k\le 3$. We prove the solvability of the problem and the compactness of the
solution sets on manifolds when $k=2$ and $3$, provided the conformal class
admits a negative $k$-admissible metric with respect to the Einstein tensor.
