Eternal Solutions to the Ricci Flow on $\R^2$

We provide the classification of eternal (or ancient) solutions of the two-dimensional Ricci flow, which is equivalent to the fast diffusion equation $\frac{\partial u}{\partial t} = \Delta \log u$ on $\R^2 \times \R.$ We show that, under the necessary assumption that for every $t \in \R$, the solution $u(\cdot, t)$ defines a complete metric of bounded curvature and bounded width, $u$ is a gradient soliton of the form $U(x,t) = \frac{2}{\beta (|x-x_0|^2 + \delta e^{2\beta t})}$, for some $x_0 \in \R^2$ and some constants $\beta >0$ and $\delta >0$.