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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 constantmean curvature $H$ embedded in $N$. This result contrasts with the theorem ofTraizet [33], who proved that every flat 3-torus admits for every positiveinteger $g$ with $g\neq 2$, connected closed embedded minimal surfaces of genus$g$ with arbitrarily large area.

Triply periodic constant mean curvature surfaces