Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

Grisha Perelman¹

In our previous paper we constructed complete solutions to the Ricci flow with surgery for arbitrary initial riemannian metric on a (closed, oriented) three-manifold [P,6.1], and used the behavior of such solutions to classify three-manifolds into three types [P,8.2]. In particular, the first type consisted of those manifolds, whose prime factors are diffeomorphic copies of spherical space forms and $\mathbb{S}^2\times\mathbb{S}^1;$ they were characterized by the property that they admit metrics, that give rise to solutions to the Ricci flow with surgery, which become extinct in finite time. While this classification was sufficient to answer topological questions, an analytical question of significant independent interest remained open, namely, whether the solution becomes extinct in finite time for every initial metric on a manifold of this type.

In this note we prove that this is indeed the case. Our argument (in conjunction with [P,§1-5]) also gives a direct proof of the so called "elliptization conjecture". It turns out that it does not require any substantially new ideas: we use only a version of the least area disk argument from [H,§11] and a regularization of the curve shortening flow from [A-G].