Eigenvalue estimates for submanifolds with locally bounded mean curvature in $N \times \mathbb {R}$

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 have a positive fundamental tone.