The mean curvature of cylindrically bounded submanifolds

Details The mean curvature of cylindrically bounded submanifolds The mean curvature of cylindrically bounded submanifolds

2008-12-03

arxiv.org/abs/0812.0623v2

Mathematische Annalen 345 (2009), 367--376.

Mathematics

Differential Geometry

First version (December 2008). Final version, including new title (February 2009). To appear in Mathematische Annalen

Scientific paper

10.1007/s00208-009-0357-1

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.

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