Graphs with parallel mean curvature

Abstract:

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 $D \subset M$, $\left \| H \right \|$ 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.