The critical point equation on a three-dimensional compact manifold

On a compact $n$-dimensional manifold $M^n$, a critical point of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation (CPE), given by $z_g=s'^*_g(f)$. It has been conjectured that a solution $(g,f)$ of the CPE is Einstein. Restricting our considerations to $n=3$ and assuming that there exist at least two distinct solutions of the CPE throughout the paper, we first prove that, if the second homology of $M^3$ vanishes, then $M^3$ is diffeomorphic to $S^3$ (Theorem 2). Secondly, we prove that the same conclusion holds if we have a lower Ricci curvature bound or the connectedness of a certain surface of $M^3$ (Theorem 3). Finally, we also prove that, if two connected surfaces of $M^3$ are disjoint, $(M^3,g)$ is isometric to a standard $3$-sphere (Theorem 4).