Eigenvalue Estimates for submanifolds of $N \times \mathbb{R}$ with   locally bounded mean curvature
  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. 
G. Pacelli Bessa, M. Silvana Costa