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- W1569181257 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 $z_g=sâ^*_g(f)$. 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)." @default.
- W1569181257 created "2016-06-24" @default.
- W1569181257 creator A5090532718 @default.
- W1569181257 date "2003-05-09" @default.
- W1569181257 modified "2023-10-18" @default.
- W1569181257 title "The critical point equation on a three-dimensional compact manifold" @default.
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- W1569181257 doi "https://doi.org/10.1090/s0002-9939-03-07165-x" @default.
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