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

Abstract

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 ${\mathbb{R}}^2\times \mathbb{R}.$ We show that, under the necessary assumption that for every $t\in \mathbb{R}$, 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 $x_0\in {\mathbb{R}}^2$ and some constants $\beta >0$ and $\delta >0$.