Minimizing Constant Mean Curvature Hypersurfaces in Hyperbolic Space

We study the constant mean curvature (CMC) hypersurfaces in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{H}^{n+1}$$\end{document} whose asymptotic boundaries are closed codimension-1 submanifolds in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^{n}_{\infty}(\mathbb{H}^{n+1})$$\end{document}. 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.