An Intrinsic Rigidity Theorem for Closed Minimal Hypersurfaces in $\mathbb{S}^5$ with Constant Nonnegative Scalar Curvature

Abstract

Let M ⁴ be a closed minimal hypersurface in $\mathbb{S}^5$ with constant nonnegative scalar curvature. Denote by f ₃ the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f ₃ and g are constant, then M ⁴ is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form of M ⁴. This result provides another piece of supporting evidence to the Chern conjecture.

Keywords

Chern conjecture / Isoparametric hypersurfaces / Scalar curvature / Minimal hypersurfaces in spheres

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