Complete CMC spacelike hypersurfaces immersed in a Lorentzian product space

We use Bochner’s formula jointly with the generalized maximum principle of Omori-Yau and an extension of Liouville’s theorem due to Yau in order to show that a complete spacelike hypersurface Σn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^{n}}$$\end{document} immersed with constant mean curvature in a Lorentzian product space M¯n+1=-R×Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{M}^{n+1}=-{\mathbb{R}}{\times}M^{n}}$$\end{document}, whose fiber Mn has nonnegative sectional curvature, must be a slice, provided that Σn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^{n}}$$\end{document} is bounded away from the future (or past) infinity of M¯n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{M}^{n+1}}$$\end{document} and that its normal hyperbolic angle is bounded. We also study the rigidity of entire vertical graphs with constant mean curvature in such an ambient space.