Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces

Abstract

In this paper we obtain a height estimate concerning compact space-like hypersurfaces Σn immersed with some positive constant r-mean curvature into an (n + 1)-dimensional Lorentzian product space \({-\mathbb{R} \times M^n}\) , and whose boundary is contained into a slice {t} × Mn. By considering the hyperbolic caps of the Lorentz–Minkowski space \({\mathbb{L}^{n+1}}\) , we show that our estimate is sharp. Furthermore, we apply this estimate to study the complete space-like hypersurfaces immersed with some positive constant r-mean curvature into a Lorentzian product space. For instance, when the ambient space–time is spatially closed, we show that such hypersurfaces must satisfy the topological property of having more than one end which constitutes a necessary condition for their existence.