关于$\boldsymbol{\text{S}^5(1)}$中具有常数量曲率的闭CMC超曲面的刚性
On the Rigidity of Closed CMC Hypersurfaces in $\mathbb{S}^5(1)$ with Constant Scalar Curvature
AI总结：
该研究证明$\boldsymbol{\text{S}^5(1)}$中具有常数量曲率的闭CMC超曲面若某点仅两个主曲率则为等参Clifford环面，加Willmore条件后可完整分类为几类等参超曲面，采用无迹张量恒等式等方法完成证明。
AI中文摘要：
设$M^4\to\boldsymbol{\text{S}^5(1)}$是一个闭CMC超曲面，具有常数量曲率和第三主曲率幂和$f_3=\boldsymbol{\text{sum}}_{i,j,k}h_{ij}h_{jk}h_{ki}$。我们证明：若$M^4$在某点处恰有两个不同的主曲率，则它是等参超曲面，具体形式为Clifford环面$\boldsymbol{\text{S}^1(r)\times\text{S}^3(\boldsymbol{\text{sqrt}}(1-r^2))}$或$\boldsymbol{\text{S}^2(r)\times\text{S}^2(\boldsymbol{\text{sqrt}}(1-r^2))}$，其中$0<r<1$。在额外的Willmore条件下，我们得到完整分类：每个闭CMC Willmore超曲面$M^4\to\boldsymbol{\text{S}^5(1)}$且具有常数量曲率的，均为等参超曲面，因此它与全脐测地球、极小Clifford环面$\boldsymbol{\text{S}^2(1/\boldsymbol{\text{sqrt}}2)\times\text{S}^2(1/\boldsymbol{\text{sqrt}}2)$、非极小Clifford环面$\boldsymbol{\text{S}^1(\boldsymbol{\text{sqrt}}3/2)\times\text{S}^3(1/2)}$或Cartan极小超曲面全等。证明结合了无迹局部张量恒等式、主曲率重数的代数分析，以及带截断论证的加权微分3-形式（截断在主曲率重合的集合附近），且未对数量曲率施加符号条件。
英文摘要：
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.
