  Given a closed flat 3-torus $N$, for each $H>0$ and each non-negative integer
$g$, we obtain area estimates for closed surfaces with genus $g$ and constant
mean curvature $H$ embedded in $N$. This result contrasts with the theorem of
Traizet [33], who proved that every flat 3-torus admits for every positive
integer $g$ with $g\neq 2$, connected closed embedded minimal surfaces of genus
$g$ with arbitrarily large area.
