Scalar curvature of minimal hypersurfaces in a sphere

We improve the well-known scalar curvature pinching theorem due to Peng–Terng for n (n⩽5)-dimensional minimal hypersurfaces to the case of arbitrary n. Precisely, if M is a closed and minimal hypersurface in a unit sphere Sn+1, then there exists a positive constant δ(n) depending only on n such that if n⩽S⩽n+δ(n), then S≡n, i.e., M is a Clifford torus , k=1,2,…,n−1.