Given a calibrated Riemannian manifold $\overline{M}$ with parallel
calibration $\Omega$ of rank $m$ and $M$ an orientable
m-submanifold with parallel mean curvature $H$, we prove that if
$\cos\theta$ is bounded away from zero, where $\theta$ is the
$\Omega$-angle of $M$, and if $M$ has zero Cheeger constant, then
$M$ is minimal. In the particular case $M$ is complete with
$Ricci^M\geq 0$ we may replace
the boundedness condition on $\cos\theta$ by $\cos\theta\geq
Cr^{-\beta}$, when $r\rightarrow+\infty$, where $0 < \beta < 1$ and $C > 0$
are constants and $r$ is the distance function to a point in $M$.
Our proof is surprisingly simple and extends to a very large class of
submanifolds in calibrated manifolds, in a unified way, the
problem started by Heinz and Chern of estimating the mean curvature
of graphic hypersurfaces in Euclidean spaces. It is based on an
estimation of $\|H\|$ in terms of $\cos\theta$ and an isoperimetric
inequality. In a similar way, we also give some conditions to
conclude $M$ is totally geodesic. We study some particular cases.