Rigidity and stability estimates for minimal submanifolds in the\n hyperbolic space

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