Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space

Let Mn be a complete non-compact submanifold in the hyperbolic space Hn+m. We first give an estimate for the bottom of the spectral of the Laplace operator on Mn, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L2 harmonic forms in certain degrees if the total mean curvature of Mn is less than an explicit constant and its total curvature is less than a suitable related constant. In addition, we obtain some vanishing results under certain pointwise restrictions on the traceless second fundamental form. Moreover, according to the nonexistence of nontrivial L2 harmonic 1-forms, we can further prove some one-end theorems.