Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. A.L. Besse, Einstein Manifolds (Springer-Verlag, New York, 1987) MR 88f:53087
  • 2. Bessières, Lafontaine and Rozoy, Scalar Curvature and Black Holes, Séminaire de Théorie Spectrale et Géométrie, Vol. 18, Année 1999-2000, 69-76, Sémin. Théor. Spectr. Géom., 18, Univ. Grenoble I, Saint-Martin-d'Hères, 2000. MR 2002b:53065
  • 3. A.E. Fischer and J.E. Marsden, Manifolds of Riemannian Metrics with Prescribed Scalar Curvature, Bull. Am. Math. Soc. 80, 479-484 (1974) MR 49:11561
  • 4. G. Galloway, On the Topology of Black Holes, Comm. Math. Phys. 151 55-66 (1993) MR 93k:83047
  • 5. S. Hwang, Critical points and conformally flat metrics, Bull. Korean Math. Soc. 37, No. 3, 641-648 (2000) MR 2001m:58033
  • 6. S. Hwang, Critical points of the scalar curvature functionals on the space of metrics of constant scalar curvature, Manuscripta Math. 103 135-142 (2000) MR 2001m:58032
  • 7. S. Hwang, A note on the circle action on Einstein manifold, Bull. Austral. Math. Soc. 63 83-91 (2001) MR 2002e:53061
  • 8. O. Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34, No. 4, 665-675 (1982) MR 84a:53046
  • 9. J. Lafontaine, Sur la géométrie d'une généralisation de l'équation différentielle d'Obata, J. Math. Pures Appliquées 62, 63-72 (1983) MR 84i:53047
  • 10. H.B. Lawson, Minimal varieties in real and complex geometry (University of Montreal lecture notes, 1974). MR 57:13798
  • 11. W. Meeks, L. Simon, and S.-T. Yau, Emedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. Math. 116, 621-659 (1982) MR 84f:53053
  • 12. 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
  • 13. Y. Shen, A note on Fisher-Marsden's conjecture, Proc. Amer. Math. Soc. 125, No.3, 901-905 (1997) MR 97k:53039

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C25

Retrieve articles in all journals with MSC (2000): 53C25

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

American Mathematical Society