A pinching theorem for minimal submanifolds in a sphere (Q1193266)

This paper deals with the pinching problem for the length square of the second fundamental form of a compact minimal submanifold in a sphere. The main results are as follows: Let \(M\) be a compact \(n\)-dimensional minimal submanifold in a unit sphere \(S^{n+p}\) of dimension \(n+p\). Denote by \(S\) the length square of the second fundamental form of \(M\). If \(n\) is even and \(S\leq n/[1+2(n-1)/(3n-2)]\), then \(M\) is either totally geodesic or a Veronese surface in \(S^ 4\). Moreover, the case where \(n\) is odd is considered similarly. Recently, a better uniform pinching constant was shown by \textit{Li Anmin} and \textit{Li Jimin} [see Arch. Math. 58, 582-594 (1992; Zbl 0731.53056)].