On the structure of submanifolds in the hyperbolic space

Let M n be a complete submanifold in the hyperbolic space H n + m . We show the vanishing of the Betti numbers β p ( M ) , 1 ≤ p ≤ n - 1 , if M is compact and the squared norm of the mean curvature satisfies some pinching condition. In the noncompact case, we prove various vanishing theorems of L q harmonic p -forms on M n if the mean curvature is bounded from above or below, and the total curvature is less than an explicit constant or some stability type inequality holds on M n . Finally, by putting some restrictions on the bottom of the spectrum of the Laplace operator, we can also get some vanishing theorems. On the other hand, based on the nonexistence of nontrivial L 2 harmonic 1-forms on M n , we can further show some one-end theorems under various hypotheses.