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 ${\mathbb{S}}^2(1/\sqrt{2})\times {\mathbb{S}}^2(1/\sqrt{2})$, the nonminimal Clifford torus ${\mathbb{S}}^1(\sqrt{3}/2)\times {\mathbb{S}}^3(1/2)$, 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.

21 pages