An estimate of the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere (Q1340224)

Let \(M^n\) \((n>3)\) be a closed minimal surface with constant scalar curvature in the unit sphere \(S^{n+1}(1)\) and \(S\) the square of the length of its second fundamental form. In this paper, we prove that \(S>n\) implies estimates of the form \(S>n+ cn-d\) with \(c\geq {1\over 4}\). For example, for \(n>17\) and \(S>n\) we prove \(S>n+{1\over 4}n\) which is sharper than a recent result of the authors [Chin. Sci. Bull. 36, 1-6 (1991; Zbl 0737.53056)].