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.
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Dedicated to Professor Manfredo P. do Carmo on the occasion of his 80th birthday.
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Alías, L.J., Bessa, G.P. & Dajczer, M. The mean curvature of cylindrically bounded submanifolds. Math. Ann. 345, 367–376 (2009). https://doi.org/10.1007/s00208-009-0357-1
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DOI: https://doi.org/10.1007/s00208-009-0357-1