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

摘要

In this paper we obtain a height estimate concerning compact space-like hypersurfaces Σ n immersed with some positive constant r-mean curvature into an (n + 1)-dimensional Lorentzian product space -mathbbR ×Mn{-mathbb{R} times M^n} , and whose boundary is contained into a slice {t} × M n . By considering the hyperbolic caps of the Lorentz–Minkowski space mathbbLn+1{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.