Classification of ancient compact solutions to the Ricci flow on surfaces

We consider an ancient solution $g(•, t)$ of the Ricci flow on a compact surface that exists for t∈ (−∞, T) and becomes spherical at time $t = T$. We prove that the metric $g(•, t)$ is either a family of contracting spheres, which is a type I ancient solution, or a King–Rosenau solution, which is a type II ancient solution.