Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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TOTAL SCALAR CURVATURE AND EXISTENCE OF STABLE MINIMAL SURFACES

  • Received : 2008.09.18
  • Accepted : 2008.10.13
  • Published : 2008.12.25
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Abstract

On a compact n-dimensional manifold M, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of volume 1, should be Einstein. The purpose of the present paper is to prove that a 3-dimensional manifold (M,g) is isometric to a standard sphere if ker $s^*_g{{\neq}}0$ and there is a lower Ricci curvature bound. We also study the structure of a compact oriented stable minimal surface in M.

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References

  1. A.L. Besse, Einstein Manifolds, New York: Springer-Verlag 1987.
  2. A.E. Fischer, J.E. Marsden, Manifolds of Riemannian Metrics with Prescribed Scalar Curvature, Bull. Am. Math. Soc. 80, 479-484 (1971).
  3. S. Hwang, Critical points of the scalar curvature functionals on the space of metrics or constant scalar curvature, manuscripta math. 103, 135-142 (2000). https://doi.org/10.1007/PL00005857
  4. S. Hwang. The critical point equation on a three dimensional compact manifold, Proc. Amer. Math. Soc. 131, 3221-3230 (2003). https://doi.org/10.1090/S0002-9939-03-07165-X
  5. S. Hwang, Stable minimal hypersufaces in a critical point equation, Commun. Korean Math. Soc. 20, 775-779 (2005). https://doi.org/10.4134/CKMS.2005.20.4.775
  6. S. Hwang, Some remarks on stable minimal surfaces in the critical point of the total scalar curvature, preprint.
  7. S. Hwang, G. Yun, J. Chang, Rigidity of the critical point equation. preprint.
  8. J. Lafontaine, Sur la geometrie d'une generalisation de l'equation differentielle d'Obata, J. Math. Pures Appliquees 62, 63-72 (1983).
  9. M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14(3), 333-310 (1962). https://doi.org/10.2969/jmsj/01430333
  10. Y. Shen, A note on Fisher-Marsden's conjecture, Proc. Amer. Math. Soc. 125(3), 901-905 (1997). https://doi.org/10.1090/S0002-9939-97-03635-6