Triply periodic constant mean curvature surfaces

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 [31], who proved that every flat 3-torus admits for every positive integer g with g≠2, connected closed embedded minimal surfaces of genus g with arbitrarily large area.