New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space

Cícero P. Aquino; Henrique F. de Lima

Archivum Mathematicum (2015)

  • Volume: 051, Issue: 4, page 201-209
  • ISSN: 0044-8753

Abstract

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In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space ℍ n + 1 , that is, complete hypersurfaces of ℍ n + 1 whose mean curvature H and normalized scalar curvature R satisfy R = a H + b for some a , b ∈ ℝ . In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of ℍ n + 1 . Furthermore, a rigidity result concerning the compact case is also given.

How to cite

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Aquino, Cícero P., and de Lima, Henrique F.. "New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space." Archivum Mathematicum 051.4 (2015): 201-209. <http://eudml.org/doc/276035>.

@article{Aquino2015,
abstract = {In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space $\mathbb \{H\}^\{n+1\}$, that is, complete hypersurfaces of $\mathbb \{H\}^\{n+1\}$ whose mean curvature $H$ and normalized scalar curvature $R$ satisfy $R=aH+b$ for some $a$, $b\in \mathbb \{R\}$. In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $\mathbb \{H\}^\{n+1\}$. Furthermore, a rigidity result concerning the compact case is also given.},
author = {Aquino, Cícero P., de Lima, Henrique F.},
journal = {Archivum Mathematicum},
keywords = {hyperbolic space; linear Weingarten hypersurfaces; totally umbilical hypersurfaces; hyperbolic cylinders},
language = {eng},
number = {4},
pages = {201-209},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space},
url = {http://eudml.org/doc/276035},
volume = {051},
year = {2015},
}

TY - JOUR
AU - Aquino, Cícero P.
AU - de Lima, Henrique F.
TI - New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space
JO - Archivum Mathematicum
PY - 2015
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 051
IS - 4
SP - 201
EP - 209
AB - In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space $\mathbb {H}^{n+1}$, that is, complete hypersurfaces of $\mathbb {H}^{n+1}$ whose mean curvature $H$ and normalized scalar curvature $R$ satisfy $R=aH+b$ for some $a$, $b\in \mathbb {R}$. In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $\mathbb {H}^{n+1}$. Furthermore, a rigidity result concerning the compact case is also given.
LA - eng
KW - hyperbolic space; linear Weingarten hypersurfaces; totally umbilical hypersurfaces; hyperbolic cylinders
UR - http://eudml.org/doc/276035
ER -

References

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