On the Rigidity of Closed CMC Hypersurfaces in $\mathbb{S}^5(1)$ with Constant Scalar Curvature

Abstract

Let $M^4\hookrightarrow\mathbb S^5(1)$ be a closed CMC hypersurface with constant scalar curvature and constant third power sum $f_3=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$. We prove that if $M^4$ has exactly two distinct principal curvatures at some point, then it is isoparametric. More precisely, it is a Clifford torus of the form $\mathbb S^1(r)\times\mathbb S^3(\sqrt{1-r^2})$ or $\mathbb S^2(r)\times\mathbb S^2(\sqrt{1-r^2})$, where $0<r<1$. Under the additional Willmore condition, we obtain a complete classification: every closed CMC Willmore hypersurface $M^4\hookrightarrow\mathbb S^5(1)$ with constant scalar curvature is isoparametric. Consequently, it is congruent to a totally umbilic geodesic sphere, the minimal Clifford torus $\mathbb S^2(1/\sqrt2)\times\mathbb S^2(1/\sqrt2)$, the nonminimal Clifford torus $\mathbb S^1(\sqrt3/2)\times\mathbb S^3(1/2)$, or a Cartan minimal hypersurface. The proofs combine trace-free local tensor identities, an algebraic analysis of the possible principal curvature multiplicities, and a weighted differential $3$-form together with a cut-off argument near the set where principal curvatures coalesce. No sign condition on the scalar curvature is imposed.