The critical point equation on a three-dimensional compact manifold

Abstract

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).