Closed Minimal Willimore Hypersurfaces of S5(1) with Constant Scalar Curvature

Abstract

We consider minimal closed hypersurfaces $M^4\subset \mathbb{S}^5(1)$ with constant scalar curvature. We prove that, if $M^4$ is additionally a Willmore hypersurface, then it is isoparametric. This gives a positive answer to the question made by Chern about the pinching of the scalar curvature for closed minimal Willmore hypersurfaces in dimension 4.