A rigidity theorem for the three dimensional critical point equation

초록

On a compact 3-dimensional manifold M-3, a critical point of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satifies the critical point equation (CPE), given by U-g(*)(f) = z(g). It has been conjectured that a solution (g, f) of CPE is Einstein. In this paper, we prove that, if CPE has two distinct solution functions and Weyl-Schouten tensor vanishes on a certain hypersurface of M-3, (M-3, g) is isometric to a standard 3-sphere.

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