# STABLE MINIMAL HYPERSURFACES IN A CRITICAL POINT EQUATION

HWang, Seung-Su

• Published : 2005.10.01
• 50 6

#### 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, satifies the critical point equation (CPE), given by $Z_g\;=\;s_g^{1\ast}(f)$. It has been conjectured that a solution (g, f) of CPE is Einstein. The purpose of the present paper is to prove that every compact stable minimal hypersurface is in a certain hypersurface of $M^n$ under an assumption that Ker($s_g^{1\ast}{\neq}0$).ﾖ⨀

#### Keywords

critical point equation;stable minimal hypersurface

#### References

1. A. L. Besse, Einstein Manifolds, Springer-Verlag, New York, 1987
2. 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
3. S. Hwang, Critical points and conformally fiat metrics, Bull. Korean Math. Soc. 37 (2000), no. 3, 641-648
4. S. Hwang, The critical point equation on a three dimensional compact manifold, Proc. Amer. Math. Soc. 131 (2003), 3221-3230 https://doi.org/10.1090/S0002-9939-03-07165-X
5. H. B. Lawson, Minimal varieties in real and complex geometry, University of Montreal lecture notes, 1974
6. S. Hwang, Critical points of the scalar curvature funetionals on the space of meirics of constant scalar curvature, Manuscripta Math. 103 (2000), 135-142 https://doi.org/10.1007/PL00005857

#### Cited by

1. TOTAL SCALAR CURVATURE AND EXISTENCE OF STABLE MINIMAL SURFACES vol.30, pp.4, 2008, https://doi.org/10.5831/HMJ.2008.30.4.677