Rigidity and stability estimates for minimal submanifolds in the hyperbolic space

In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold $M^n$ in the hyperbolic space ${\mathbb{H}}^{n+m}$ in order to show that $M^n$ is totally geodesic. We also obtain sharp upper bounds estimates for the first eigenvalue of the super stability operator in the case of $M$ is a surface in ${\mathbb{H}}^4$.