Spacelike Hypersurfaces with Constant Mean Curvature in the Steady State Space

Abstract

In this paper we obtain height estimates concerning to a compact spacelike hypersurface $\Sigma^n$ immersed with constant mean curvature $H$ in the Steady State space $\mathcal H^{n+1}$, when its boundary is contained into some hyperplane of this spacetime. As a first application of these results, when $\Sigma^n$ has spherical boundary, we establish relations between its height and the radius of its boundary. Moreover, under a certain restriction on the Gauss map of $\Sigma^n$, we obtain a sharp estimate for $H$. Finally, we also apply our estimates to describe the end of a complete spacelike hypersurface and to get theorems of characterization concerning to spacelike hyperplanes in $\mathcal H^{n+1}$.