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The critical point equation on a three-dimensional compact manifold

Author: Seungsu Hwang
Journal: Proc. Amer. Math. Soc. 131 (2003), 3221-3230
MSC (2000): Primary 53C25
Published electronically: May 9, 2003
MathSciNet review: 1992863
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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).

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Additional Information

Seungsu Hwang
Affiliation: Department of Mathematics, Chung-Ang University, 221, HukSuk-Dong, DongJak-Ku, Seoul, Korea 156-756

Keywords: Total scalar curvature functional, critical point, Einstein metric, Fisher-Marsden conjecture
Received by editor(s): February 15, 2001
Published electronically: May 9, 2003
Communicated by: Bennett Chow
Article copyright: © Copyright 2003 American Mathematical Society

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