On complete cohomogeneity one hypersurfaces in the hyperbolic space

We study isometric immersions f\colon M^{n} \rightarrow \mathbb{H}^{n+1} into hyperbolic space of dimension n+1 of a complete Riemannian manifold of dimension n on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We provide a characterization if either n \geq 3 and M^{n} is compact, or n \geq 5 and the connected components of the set where the sectional curvature is constant and equal to -1 are bounded.