We obtain sharp estimates involving the mean curvatures of higher order of a complete bounded hypersurface immersed in a complete Riemannian manifold. Similar results are also given for complete spacelike hypersurfaces in Lorentzian ambient spaces.
We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)timesR^{ell}$ in a product Riemannian manifold $N^{n-ell}timesR^{ell}$. It follows that a complete hypersurface of given constant mean curvature l
ying 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.
