The mean curvature of cylindrically bounded submanifolds

Abstract

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder \({B(r)\times{\mathbb R}^{\ell}}\) in a product Riemannian manifold \({N^{n-\ell}\times{\mathbb R}^{\ell}}\) . 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 Calabi on 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.