Classification of closed minimal hypersurfaces with constant scalar curvature in $\mathbb{S}^5$

TL;DR

This paper classifies closed minimal hypersurfaces with constant scalar curvature in the 5-sphere, showing they are isoparametric with specific geometric forms, thus supporting Chern's conjecture.

Contribution

It proves that such hypersurfaces are necessarily isoparametric and identifies their specific geometric types, a significant step in understanding minimal hypersurfaces in spheres.

Findings

Hypersurfaces are either an equatorial 4-sphere, a product of spheres, or a Cartan's minimal hypersurface.

The squared norm of the second fundamental form S can only be 0, 4, or 12.

Results support Chern's conjecture on minimal hypersurfaces.

Abstract

In this paper, we prove that any closed minimal hypersurface $M^4$ in the $5$-dimensional unit sphere $\mathbb{S}^5$ with constant scalar curvature and constant $3$-th mean curvature must be isoparametric. To be precise, $M^4$ is either an equatorial 4-sphere, a product of spheres $\mathbb{S} ^{2}(\frac{\sqrt{2}}{2}) \times \mathbb{S} ^{2}(\frac{\sqrt{2}}{2})$ or $\mathbb{S} ^{1}(\frac{1}{2}) \times \mathbb{S} ^{3}(\frac{\sqrt{3}}{2})$, or a Cartan's minimal hypersurface. In particular, the value of the squared norm of the second fundamental form $S$ can only be 0, 4, or 12. This result strongly supports Chern's conjecture.