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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Eigenvalue estimates for submanifolds with locally bounded mean curvature in $ N \times \mathbb{R}$


Authors: G. Pacelli Bessa and M. Silvana Costa
Journal: Proc. Amer. Math. Soc. 137 (2009), 1093-1102
MSC (2000): Primary 53C40, 53C42; Secondary 58C40
Published electronically: October 21, 2008
MathSciNet review: 2457451
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Abstract: 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 have a positive fundamental tone.


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Additional Information

G. Pacelli Bessa
Affiliation: The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy
Address at time of publication: Department of Mathematics, Universidade Federal do Ceara-UFC, Campus do Pici, 60455-760 Fortaleza-CE, Brazil
Email: bessa@mat.ufc.br

M. Silvana Costa
Affiliation: Department of Engineering, Universidade Federal do Ceara-UFC, Campus Cariri, Av. Castelo Branco, 150, 60030-200 Juazeiro do Norte-CE, Brazil
Email: silvana_math@yahoo.com.br

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09680-9
PII: S 0002-9939(08)09680-9
Keywords: Fundamental tone estimates, minimal submanifolds, submanifolds with locally bounded mean curvature in $N\times \mathbb {R}$.
Received by editor(s): April 29, 2008
Published electronically: October 21, 2008
Additional Notes: The first author was partially supported by a CNPq-grant and ICTP Associate Schemes.
The second author was partially supported by a CNPq-scholarship
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2008 American Mathematical Society