  We give lower bounds for the fundamental tone of open sets in submanifolds
with locally bounded mean curvature in $ N \times \mathbb{R}$, where $N$ is an
$n$-dimensional complete Riemannian manifold with radial sectional curvature
$K_{N} \leq \kappa$. When the immersion is minimal our estimates are sharp. We
also show that cylindrically bounded minimal surfaces has positive fundamental
tone.
