The influence of the boundary behaviour on hypersurfaces with constant mean curvature in \(H^{n+1}\) (Q1819756)

The asymptotic boundary of complete H-hypersurfaces in the hyperbolic space is studied. There are results of the following type: If \(H\in [0,1)\) then there is an a priori lower bound for the distance between components of the asymptotic boundary, which is attained only for rotational hypersurfaces of spherical type. For \(H\neq 1\) the latter are also characterized by the fact that its boundary consists of two disjoint (n-1)-spheres, and M is regular at infinity.