  In this paper, we prove the nonexistence of $L^2$ harmonic 1-forms on a
complete super stable minimal submanifold $M$ in hyperbolic space under the
assumption that the first eigenvalue $\lambda_1 (M)$ for the Laplace operator
on $M$ is bounded below by $(2n-1)(n-1)$. Moreover, we provide sufficient
conditions for minimal submanifolds in hyperbolic space to be super stable.
