A note on the Chern Conjecture in dimension four

TL;DR

This paper investigates conditions under which a closed minimal hypersurface in a 5-sphere is isoparametric, focusing on the relationships between curvature measures and the Chern conjecture in four dimensions.

Contribution

It establishes new curvature conditions that guarantee a minimal hypersurface in -sphere is isoparametric, advancing understanding of the Chern conjecture in dimension four.

Findings

Hypersurfaces with bounded 3-mean curvature are isoparametric.

Curvature bounds involving the Gauss-Kronecker curvature imply isoparametricity.

Results support the Chern conjecture in specific geometric settings.

Abstract

Let $M^4$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ and constant 3-mean curvature $H_3$ in $\mathbb{S}^{5}$. If $H_3^2\leq \frac{1}{2}.$ and Gauss-Kronecker curvature $K_M$ satisfies $K_M\leq1$ $($or $ K_M\leq\frac{S^2}{144}$$)$, then $M^4$ is isoparametric.