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 $(M,t^{-1}g(t))$ is a point in the Euclidean and Nil cases, and a metric circle or a compact interval of positive length in the Sol case. In all three cases, the blowdown flows lifted to the universal cover, $s^{-1}\tilde{g}(s\tau )$, converge, in the pointed Cheeger–Hamilton sense, to an explicit homogeneous expanding Ricci soliton