Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space

Let M be an n-dimensional complete non-compact submanifold in a hyperbolic space with the norm of its mean curvature vector bounded by a constant alpha < n - 1. We prove in this paper that <lambda>(1) (M) greater than or equal to 1/4 (n - 1 - alpha)(2) > 0. In particular when M is minimal we have lambda (1) (M) greater than or equal to 1/4 (n - 1)(2) and this is sharp because equality holds when M is totally geodesic.