Warped Tori with Almost Non-Negative Scalar Curvature

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j\ge -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrinsic Flat sense to a flat $3$-torus.