Spacelike graphs with parallel mean curvature

Abstract

We consider spacelike graphs $\Gamma_f$ of simple products $(M\times N, g\times -h)$ where $(M,g)$ and $(N,h)$ are Riemannian manifolds and $f:M\to N$ is a smooth map. Under the condition of the Cheeger constant of $M$ to be zero and some condition on the second fundamental form at infinity, we conclude that if $\Gamma_f \subset M\times N$ has parallel mean curvature $H$ then $H=0$. This holds trivially if $M$ is closed. If $M$ is the $m$-hyperbolic space then for any constant $c$, we describe a explicit foliation of $H^m\times R$ by hypersurfaces with constant mean curvature $c$.