The mean curvature of cylindrically bounded submanifolds

TL;DR

This paper provides estimates for the mean curvature of complete submanifolds within cylindrical regions in product manifolds, with implications for minimal hypersurfaces and their properness.

Contribution

It introduces new bounds on mean curvature for submanifolds in cylindrical regions, impacting the understanding of minimal hypersurfaces and their properness in Euclidean space.

Findings

Complete hypersurfaces with constant mean curvature cannot be proper in small cylinders.

Counterexamples to certain minimal hypersurface conjectures cannot be proper.

Submanifolds with small mean curvature are shown to be stochastically incomplete.

Abstract

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.