The critical point equation on a three-dimensional compact manifold
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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).References
- Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
- Jacques Lafontaine and Luc Rozoy, Courbure scalaire et trous noirs, Séminaire de Théorie Spectrale et Géométrie, Vol. 18, Année 1999–2000, Sémin. Théor. Spectr. Géom., vol. 18, Univ. Grenoble I, Saint-Martin-d’Hères, 2000, pp. 69–76 (French). MR 1812213, DOI 10.5802/tsg.223
- Arthur E. Fischer and Jerrold E. Marsden, Manifolds of Riemannian metrics with prescribed scalar curvature, Bull. Amer. Math. Soc. 80 (1974), 479–484. MR 346839, DOI 10.1090/S0002-9904-1974-13457-9
- Gregory J. Galloway, On the topology of black holes, Comm. Math. Phys. 151 (1993), no. 1, 53–66. MR 1201655
- Seungsu Hwang, Critical points and conformally flat metrics, Bull. Korean Math. Soc. 37 (2000), no. 3, 641–648. MR 1779252
- Seungsu Hwang, Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature, Manuscripta Math. 103 (2000), no. 2, 135–142. MR 1796310, DOI 10.1007/PL00005857
- Seungsu Hwang, A note on the circle actions on Einstein manifolds, Bull. Austral. Math. Soc. 63 (2001), no. 1, 83–91. MR 1812311, DOI 10.1017/S0004972700019134
- Osamu Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34 (1982), no. 4, 665–675. MR 669275, DOI 10.2969/jmsj/03440665
- Jacques Lafontaine, Sur la géométrie d’une généralisation de l’équation différentielle d’Obata, J. Math. Pures Appl. (9) 62 (1983), no. 1, 63–72 (French). MR 700048
- H. Blaine Lawson Jr., Minimal varieties in real and complex geometry, Séminaire de Mathématiques Supérieures, No. 57 (Été 1973), Les Presses de l’Université de Montréal, Montreal, Que., 1974. MR 0474148
- William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. MR 678484, DOI 10.2307/2007026
- M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14, No. 3, 333-340 (1962) MR25:5479
- Ying Shen, A note on Fischer-Marsden’s conjecture, Proc. Amer. Math. Soc. 125 (1997), no. 3, 901–905. MR 1353399, DOI 10.1090/S0002-9939-97-03635-6
Additional Information
- Seungsu Hwang
- Affiliation: Department of Mathematics, Chung-Ang University, 221, HukSuk-Dong, DongJak-Ku, Seoul, Korea 156-756
- Email: seungsu@cau.ac.kr
- Received by editor(s): February 15, 2001
- Published electronically: May 9, 2003
- Communicated by: Bennett Chow
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3221-3230
- MSC (2000): Primary 53C25
- DOI: https://doi.org/10.1090/S0002-9939-03-07165-X
- MathSciNet review: 1992863