Closed Minimal Hypersurfaces in ${\mathbb{S}}^5(1)$ with Constant Scalar and Gauss-Kronecker Curvatures

Abstract

In this paper, we prove that any closed minimal hypersurface $M^4$ of ${\mathbb{S}}^5(1)$ with constant scalar curvature and constant Gauss-Kronecker curvature must be isoparametric. Specifically, $M^4$ is either an equatorial 4-sphere, a Clifford torus ${\mathbb{S}}^2\left(\frac{\sqrt{2}}{2}\right)\times {\mathbb{S}}^2\left(\frac{\sqrt{2}}{2}\right)$ or ${\mathbb{S}}^1\left(\frac{1}{2}\right)\times {\mathbb{S}}^3\left(\frac{\sqrt{3}}{2}\right)$, or a Cartan's minimal hypersurface. Consequently, the squared norm of the second fundamental form $S$ can only take the values 0, 4, 12. This result provides strong support for Chern's Conjecture.

34 pages main text, 8 pages appendix