Rigidity of complete minimal submanifolds in a hyperbolic space

In this paper we prove some gap theorem for complete immersed minimal submanifold of dimension no less than six or four, depending on the codimension, in a hyperbolic space $$\mathbb {H}^{n+m}(-1)$$ . That is, we show that a high dimensional complete immersed minimal submanifold M in $$ \mathbb {H}^{n+m}(-1)$$ , is totally geodesic if the $$L^d$$ norm of |A|, for some d, on geodesic balls centered at some point $$p \in M $$ has less than quadratic growth and if either $$\sup _{x \in M} |A|^2$$ is not too large or the $$L^n$$ norm of |A| on M is finite, were, A is the second fundamental form of M.