Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THREE DIMENSIONAL CRITICAL POINT OF THE TOTAL SCALAR CURVATURE

  • Received : 2012.03.26
  • Published : 2013.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

It has been conjectured that, on a compact 3-dimensional orientable manifold, a critical point of the total scalar curvature restricted to the space of constant scalar curvature metrics of unit volume is Einstein. In this paper we prove this conjecture under a condition that ker $s^{\prime}^*_g{\neq}0$, which generalizes the previous partial results.

Keywords

References

  1. A. L. Besse, Einstein Manifolds, New York, Springer-Verlag, 1987.
  2. J. Chang, G. Yun, and S. Hwang, Critical point metrics of the total scalar curvature, Bull. Korean Math. Soc. 49 (2012), no 3. 655-667. https://doi.org/10.4134/BKMS.2012.49.3.655
  3. A. E. Fischer and J. E. Marsden, Manifolds of Riemannian metrics with prescribed scalar curvature, Bull. Amer. Math. Soc. 80 (1974), 479-484. https://doi.org/10.1090/S0002-9904-1974-13457-9
  4. S. Hwang, The critical point equation on a three dimensional compact manifold, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3221-3230. https://doi.org/10.1090/S0002-9939-03-07165-X
  5. S. Hwang, Some remarks on stable minimal surfaces in the critical point of the total scalar curvature, Comm. Korean Math. Soc. 23 (2008), no. 4, 587-595. https://doi.org/10.4134/CKMS.2008.23.4.587
  6. S. Hwang, J. Chang, and G. Yun, Rigidity of the critical point equation, Math. Nachr. 283 (2010), no. 6, 846-853. https://doi.org/10.1002/mana.200710037
  7. M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), no. 3, 333-340. https://doi.org/10.2969/jmsj/01430333

Cited by

  1. A note on critical point metrics of the total scalar curvature functional vol.424, pp.2, 2015, https://doi.org/10.1016/j.jmaa.2014.11.040
  2. Brown–York mass and positive scalar curvature II: Besse’s conjecture and related problems pp.1572-9060, 2019, https://doi.org/10.1007/s10455-019-09653-0