Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지

TOPOLOGICAL ASPECTS OF THE THREE DIMENSIONAL CRITICAL POINT EQUATION

  • CHANG, JEONGWOOK (Department of Mathematics Kunsan National University)
  • Received : 2005.08.10
  • Accepted : 2005.09.06
  • Published : 2005.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

Let ($M^n$, g) be a compact oriented Riemannian manifold. It has been conjectured that every solution of the equation $z_g=D_gdf-{\Delta}_gfg-fr_g$ is an Einstein metric. In this article, we deal with the 3 dimensional case of the equation. In dimension 3, if the conjecture fails, there should be a stable minimal hypersurface in ($M^3$, g). We study some necessary conditions to guarantee that a stable minimal hypersurface exists in $M^3$.

Keywords

References

  1. Einstein Manifolds Besse, A.L.
  2. Bulletin of the AMS v.80 Manifolds of Riemannian Metrics with Prescribed Scalar Curvature Fischer, A.E.;Marsden, J.E.
  3. Procedings of the AMS v.131 no.10 The critical point equation on a three-dimensional compact manifold Hwang, S.
  4. Stable minimal hypersurfaces in a critical point equation Hwang, S.
  5. University of Montreal lecture notes Minimal varieties in real and complex geometry Lawson, H.B.
  6. J. Math. Soc. Japan v.14 no.3 Certain conditions for a Riemannian manifold to be isometric with a sphere Obata, M.