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We study the uniqueness of horospheres and equidistant spheres in hyperbolicspace under different conditions. First we generalize the Bernstein theorem byDo Carmo and Lawson to the embedded hypersurfaces with constant higher ordermean curvature. Then we prove two Bernstein type results for immersedhypersurfaces under different assumptions. Last, we show the rigidity ofhorospheres and equidistant spheres in terms of their higher order meancurvatures.

Uniqueness of Hypersurfaces of Constant Higher Order Mean Curvature in Hyperbolic Space