Under assumptions on principal curvatures and constant S, f3, f4, closed minimal hypersurfaces M^5 in S^6 are isoparametric; those with a point of exactly two distinct principal curvatures are Clifford tori.
On Chern's conjecture for minimal hypersurfaces in spheres
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abstract
Using a new estimate for the Peng-Terng invariant and the multiple-parameter method, we verify a rigidity theorem on the stronger version of Chern Conjecture for minimal hypersurfaces in spheres. More precisely, we prove that if $M$ is a compact minimal hypersurface in $\mathbb{S}^{n+1}$ whose squared length of the second fundamental form satisfies $0\leq S-n\leq\frac{n}{18}$, then $S\equiv n$ and $M$ is a Clifford torus.
fields
math.DG 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Minimal Hypersurfaces with constant scalar curvature in $\mathbf{S}^6$
Under assumptions on principal curvatures and constant S, f3, f4, closed minimal hypersurfaces M^5 in S^6 are isoparametric; those with a point of exactly two distinct principal curvatures are Clifford tori.