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

Abstract

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 α < n-1. We prove in this paper that λ1 (M) ≥ 1/4 (n - 1 - α)2 > 0. In particular when M is minimal we have λ1 (M) ≥ 1/4 (n - 1)2 and this is sharp because equality holds when M is totally geodesic.