On the geometry of linear Weingarten hypersurfaces in the hyperbolic space

In this paper, we apply some forms of generalized maximum principles in order to study the geometry of complete linear Weingarten hypersurfaces with nonnegative sectional curvature immersed in the hyperbolic space. In this setting, under the assumption that the mean curvature attains its maximum, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder.