Graphs with parallel mean curvature

We prove that if the graph ${\Gamma _f} = \left\{ {\left( {x,f\left( x \right)} \right):x \in M} \right\}$ of a map $f:\left( {M,g} \right) \to \left( {N,h} \right)$ between Riemannian manifolds is a submanifold of $\left( {M \times N,g \times h} \right)$ with parallel mean curvature $H$, then on a compact domain $D \subset M$, $\left\Vert H \right\Vert$ is bounded from above by $\frac{1}{m}\frac{{A\left( {\partial D} \right)}}{{V\left( D \right)}}$. In particular, ${\Gamma _f}$ is minimal provided $M$ is compact, or noncompact with zero Cheeger constant. Moreover, if $M$ is the $m$-hyperbolic space--thus with nonzero Cheeger constant--then there exist real-valued functions the graphs of which are nonminimal submanifolds of $M \times \mathbb{R}$ with parallel mean curvature.