  We study the uniqueness of horospheres and equidistant spheres in hyperbolic
space under different conditions. First we generalize the Bernstein theorem by
Do Carmo and Lawson to the embedded hypersurfaces with constant higher order
mean curvature. Then we prove two Bernstein type results for immersed
hypersurfaces under different assumptions. Last, we show the rigidity of
horospheres and equidistant spheres in terms of their higher order mean
curvatures.
