An intrinsic rigidity theorem for closed minimal hypersurfaces in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature

TL;DR

This paper proves a rigidity theorem for closed minimal hypersurfaces in 5-spheres with constant nonnegative scalar curvature, showing they are isoparametric under specific curvature conditions, and classifies their second fundamental form.

Contribution

It establishes a new intrinsic rigidity result for minimal hypersurfaces in 5-spheres, supporting the Chern conjecture with explicit curvature classifications.

Findings

Hypersurfaces are isoparametric under given conditions

All possible squared lengths of the second fundamental form are classified

Provides evidence supporting the Chern conjecture

Abstract

Let M be a closed minimal hypersurface in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature. We prove that, if the sum of the cubes of all principal curvatures and the number of distinct principal curvatures are constant, then M is isoparametric. Moreover, We give all possible values for squared length of the second fundamental form of M. This result provides another piece of supporting evidence to the Chern conjecture.