Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe

In this study, we deal with non -degenerate translators of the mean curvature flow in the wellknown hyperbolic Einstein's static universe. We classify translators foliated by horospheres and rotationally invariant ones, both space -like and time -like. For space -like translators, we show a uniqueness theorem as well as a result to extend an isometry of the boundary of the domain to the whole translator, under simple conditions. As an application, we obtain a characterization of the the bowl when the boundary is a ball, and of certain translators foliated by horospheres whose boundary is a rectangle.