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The non-parabolicity of infinite volume ends


Authors: M. P. Cavalcante, H. Mirandola and F. Vitório
Journal: Proc. Amer. Math. Soc. 143 (2015), 1221-1228
MSC (2010): Primary 53C40; Secondary 53C20
Published electronically: November 20, 2014
MathSciNet review: 3293737
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Abstract: Let $ M^m$, with $ m\geq 3$, be an $ m$-dimensional complete non-compact manifold isometrically immersed in a Hadamard manifold $ \bar M$. Assume that the mean curvature vector has finite $ L^p$-norm, for some $ 2\leq p\leq m$. We prove that each end of $ M$ must either have finite volume or be non-parabolic.


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Additional Information

M. P. Cavalcante
Affiliation: Instituto de Matemática, Universidade Federal de Alagoas, Maceió, AL, CEP 57072-970, Brazil
Email: marcos.petrucio@pq.cnpq.br

H. Mirandola
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, CEP 21945-970, Brasil
Email: mirandola@im.ufrj.br

F. Vitório
Affiliation: Instituto de Matemática, Universidade Federal de Alagoas, Maceió, AL, CEP 57072-970, Brazil
Email: feliciano.vitorio@pq.cnpq.br

DOI: https://doi.org/10.1090/S0002-9939-2014-11901-0
Keywords: Parabolicity, Sobolev inequalities, ends, total mean curvature.
Received by editor(s): February 4, 2012
Received by editor(s) in revised form: April 17, 2013
Published electronically: November 20, 2014
Additional Notes: The first and third authors were partially supported by CNPq under the grants 483268/2010-0
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.