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Register a new userEigenvalue Estimates for submanifolds of $N times mathbb{R}$ with locally bounded mean curvature
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Greg\\'orio Pacelli F. Bessa
Publication date
2008
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and research's language is
English
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We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $ N times mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} leq kappa$. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces has positive fundamental tone.
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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 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.
Jorge-Koutrofiotis and Pigola-Rigoli-Setti proved sharp sectional curvature estimates for extrinsically bounded submanifolds. Alias, Bessa and Montenegro showed that these estimates hold on properly immersed cylindrically bounded submanifolds. On the other hand, Alias, Bessa and Dajczer proved sharp mean curvature estimates for properly immersed cylindrically bounded submanifolds. In this paper we prove these sectional and mean curvature estimates for a larger class of submanifolds, the properly immersed $phi$-bounded submanifolds.
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 lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type $N^n times_f Q^q$, where $f in C^infty(N)$. We also study the essential spectrum of these minimal submanifolds.
Based on Markvorsen and Palmers work on mean time exit and isoperimetric inequalities we establish slightly better isoperimetric inequalities and mean time exit estimates for minimal submanifolds of $Ntimesmathbb{R}$. We also prove isoperimetric inequalities for submanifolds of Hadamard spaces with tamed second fundamental form.
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