We give an estimate of the mean curvature of a complete submanifold lying
inside a closed cylinder B(r)×Rℓ in a product Riemannian manifold
Nn−ℓ×Rℓ. 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.Comment: First version (December 2008). Final version, including new title
(February 2009). To appear in Mathematische Annale