Rigidity of Complete Minimal Submanifolds in Spheres - Zhou, Jundong (Frontiers in Physics, 2020)
Let $M$ be an $n$-dimensional complete minimal submanifold in an $(n+p)$-dimensional sphere $S^{n+p}$ and $h$ be the second fundamental form of $M$. In this paper, it is shown that $M$ is totally geodesic if the $L^2$ norm of $|h|$ on any geodesic ball of $M$ has less than quadratic growth and $L^n$ norm of $|h|$ on $M$ is less than a fixed constant. Farther, deleting the condition of the $L^2$ norm of $|h|$, we show that $M$ is totally geodesic if $L^n$ norm of $|h|$ on $M$ is less than a fixed constant. Furthermore, we provide a sufficient condition such that a complete stable minimal hypersurface is totally geodesic.