Under Ricci flow, a 3-torus goes flat

We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type $\mathbb{R}^3$, Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric $g(t)$ converges exponentially fast to a flat metric. The Gromov--Hausdorff limit of $