Some Rigidity Results of Minimal Submanifolds in Space Forms

In this paper, we consider some rigidity theorems for minimal submanifolds of space forms. Firstly, by introducing a new parameter in the process of proof, we improve a rigidity result of Chen-Wei for minimal closed submanifolds of the unit sphere under the integral Ricci curvature condition. On the other hand, for a complete minimal submanifold of a hyperbolic space, we show that if the dimension of the submanifold is not less than 4 and the second fundamental form is bounded from above by a constant depending only on the dimension, then the submanifold must be totally geodesic under an integral condition with respect to the second fundamental form.