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Keywords:
hyperbolic space; linear Weingarten hypersurfaces; totally umbilical hypersurfaces; hyperbolic cylinders
hyperbolic space; linear Weingarten hypersurfaces; totally umbilical hypersurfaces; hyperbolic cylinders
Summary:
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.
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