The critical point equation on a three-dimensional compact manifold

초록

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).

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