Geodesic
New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space
Archivum mathematicum, Tome 51 (2015) no. 4, pp. 201-209

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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.
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.
DOI : 10.5817/AM2015-4-201
Classification : 53A10, 53B30, 53C42, 53C50
Keywords: hyperbolic space; linear Weingarten hypersurfaces; totally umbilical hypersurfaces; hyperbolic cylinders 
Aquino, Cícero P.; de Lima, Henrique F. New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space. Archivum mathematicum, Tome 51 (2015) no. 4, pp. 201-209. doi: 10.5817/AM2015-4-201
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     mrnumber = {3434603},
     zbl = {06537725},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2015-4-201/}
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