Minimal submanifolds of a sphere with bounded second fundamental form

Let $h$ be the second fundamental form of an $n$-dimensional minimal submanifold $M$ of a unit sphere ${S^{n + p}}(p \geqslant 2)$, $S$ be the square of the length of $h$, and $\sigma (u) = \vert\vert h(u,u)\vert{\vert^2}$ for any unit vector $u \in TM$. Simons proved that if $S \leqslant n/(2 - 1/p)$ on $M$, then either $S \equiv 0$, or $S \equiv n/(2 - 1/p)$. Chern, do Carmo, and Kobayashi determined all minimal submanifolds satisfying $S \equiv n/(2 - 1/p)$. In this paper the analogous results for $\sigma (u)$ are obtained. It is proved that if $\sigma (u) \leqslant \tfrac{1} {3}$, then either $\sigma (u) \equiv 0$, or $\sigma (u) \equiv \tfrac{1} {3}$. All minimal submanifolds satisfying $\sigma (u)$ are determined. A stronger result is obtained if $M$ is odd-dimensional.