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.