《Finite extinction time for the solutions to the Ricci flow on certain three-manifolds》.pdf

3 Finite extinction time for the solutions to the 0 0 Ricci flow on certain three-manifolds 2 l Grisha Perelman∗ u J 7 February 1, 2008 1 ] G In our previous paper we constructed complete solutions to the Ricci flow D 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- h manifolds into three types [P,8.2]. In particular, the first type consisted of those t a manifolds, whose prime factors are diffeomorphic copies of spherical space forms 2 1 m and S × S ; 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 ques- 1 tions, an analytical question of significant independent interest remained open, v 5 namely, whether the solution becomes extinct in finite time for every initial 4 metric on a manifold of this type. 2 In this note we prove that this is indeed the case. Our argument (in con- 7 junction with [P, §1-5]) also gives a direct proof of the so called ”elliptization 0 conjecture”. It turns out that it does not require any substantially new ideas: 3 0 we use only a version of the least area disk argument from [H,§11] and a regu- /