Rigidity of compact minimal submanifolds in a unit sphere
The authors prove the following result: Let \(M\) be a smooth \(n\)- dimensional compact minimal submanifold of a unit sphere of codimension \(p\geq 2\), and denote by \(S\) the square of the length of the second fundamental form of this submanifold. If \(S\leq{2\over 3}n\) on \(M\), then \(M\) is totally geodesic or a Veronese surface. This result was already proved by \textit{Li An-Min} and \textit{Li Jimin} [Arch. Math. 58, 582-594 (1992)]. All authors use the formula for \(S\) of \textit{J. Simons} [Ann. Math., II. Ser. 88, 62-105 (1968; Zbl 0181.497)], but give different proofs for some subtil estimates.
scientific article; zbMATH DE number 149052