Rigidity of Closed Minimal Hypersurfaces in $\mathbb{S}^5$

Abstract

The celebrated Chern conjecture asserts that any closed minimal hypersurface in $\mathbb{S}^{n+1}$ with constant scalar curvature is isoparametric. In this paper, we resolve this conjecture in the affirmative for $M^4 \subset \mathbb S^5$ under the assumption that the Gauss-Kronecker curvature $K$ is constant. This result breaks the traditional reliance on consecutive trace conditions, demonstrating that the nonconsecutive spectral invariant set $\{H, S, K\}$ is sufficient to yield complete geometric rigidity. To overcome the analytical singular locus, we construct two novel weighted $3$-forms adapted to $S$ and $K$. Crucially, the global curvature estimates required to close our analysis are obtained unconditionally by proving the Euler characteristic $\chi(M)=0$. This local-to-global approach provides a new paradigm for higher-dimensional rigidity problems.