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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An estimate for the sectional curvature of cylindrically bounded submanifolds


Authors: Luis J. Alías, G. Pacelli Bessa and J. Fabio Montenegro
Journal: Trans. Amer. Math. Soc. 364 (2012), 3513-3528
MSC (2010): Primary 53C42, 53C40
Published electronically: February 27, 2012
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Abstract: We give sharp sectional curvature estimates for complete immersed cylindrically bounded $ m$-submanifolds $ \varphi \colon M^m \to N^{n-\ell }\times \mathbb{R}^{\ell }$, $ n+\ell \leq 2m-1$, provided that either $ \varphi $ is proper with the norm of the second fundamental form with certain controlled growth or $ M$ has scalar curvature with strong quadratic decay. The latter gives a non-trivial extension of the Jorge-Koutrofiotis Theorem. In the particular case of hypersurfaces, that is, $ m=n-1$, the growth rate of the norm of the second fundamental form is improved. Our results will be an application of a generalized Omori-Yau Maximum Principle for the Hessian of a Riemannian manifold, in its newest elaboration given by Pigola, Rigoli and Setti (2005).


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Additional Information

Luis J. Alías
Affiliation: Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain
Email: ljalias@um.es

G. Pacelli Bessa
Affiliation: Departamento de Matemática, Universidade Federal do Ceara-UFC, Bloco 914, Campus do Pici, 60455-760, Fortaleza, Ceara, Brazil
Email: bessa@mat.ufc.br

J. Fabio Montenegro
Affiliation: Departamento de Matemática, Universidade Federal do Ceara-UFC, Bloco 914, Campus do Pici, 60455-760, Fortaleza, Ceara, Brazil
Email: fabio@mat.ufc.br

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05439-0
PII: S 0002-9947(2012)05439-0
Keywords: Omori-Yau Maximum Principle, cylindrically bounded submanifolds, properly immersed submanifolds.
Received by editor(s): June 11, 2010
Published electronically: February 27, 2012
Additional Notes: The first author’s research was a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007-2010). His research was partially supported by MEC project PCI2006-A7-0532, MICINN project MTM2009-10418, and Fundación Séneca project 04540/GERM/06, Spain
The second author’s research was partially supported by CNPq-CAPES (Brazil) and MEC project PCI2006-A7-0532 (Spain)
The third author’s research was partially supported by CNPq-CAPES (Brazil).
Article copyright: © Copyright 2012 American Mathematical Society