Stability for a class of three-tori with small negative scalar curvature

We define a flexible class of Riemmanian metrics on the three-torus. Then, using Stern's inequality relating scalar curvature to harmonic one-forms, we show that any sequence of metrics in this family whose negative part of the scalar curvature tends to zero in $L^2$ norm has a subsequence which converges to some flat metric on the three-torus in the sense of Dong-Song.