Closed Willmore Minimal Hypersurfaces With Constant Scalar Curvature In S~5(1) Are Isoparametric

Minimal hypersurfaces in unit spheres are important in submanifold geometry, and the Chern’s conjecture is an important problem about it. In 1968, Chern con-jectured that:let S be the value of the squared norm of the second fundamental form for n-dimensional closed minimal hypersurfaces in the unit sphere Sn+1 with constant scalar curvature, then the set of S should be discrete. After the efforts of many mathematicians, S. P. Chang had completely solved the case n = 3 in 1994. For higher dimensional cases, the problem is still unsolved.For n=4, T. Lusada, M. Scherfner and L. A. M. Sousa. Jr considered the closed Willmore minimal hypersurface in S5 in 2005. Under the condition of nonnegative constant scalar curvature, they proved it must be isoparametric (see [12]), which strongly supported the Chern’s conjecture. Removing the nonnegative condition, we prove that:any closed minimal Willmore hypersurfaces M4 of S5(1) with constant scalar curvature must be isoparametric. To be precise, M4 is either an equatorial 4 sphere, a product of sphere S2((?)/2)×S2((?)/2) or a Cartan’s minimal hypersurface, in particular, S can only be 0,4,12, which better supports the Chern’s conjecture.