Minimizing constant mean curvature hypersurfaces in hyperbolic space

We study the constant mean curvature (CMC) hypersurfaces in Hn+1 whose asymptotic boundaries are closed codimension-1 submanifolds in S-infinity(n)(Hn+1). We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area-minimizing, convex hull property, exchange roundoff trick to CMC hypersurface context. We also give a generic uniqueness result for CMC hypersurfaces in hyperbolic space.