  We give an estimate of the mean curvature of a complete submanifold lying
inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold
$N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given
constant mean curvature lying inside a closed circular cylinder in Euclidean
space cannot be proper if the circular base is of sufficiently small radius. In
particular, any possible counterexample to a conjecture of Calabion complete
minimal hypersurfaces cannot be proper. As another application of our method,
we derive a result about the stochastic incompleteness of submanifolds with
sufficiently small mean curvature.
