  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 $\mathbb{H}^{4}$.
