Ricci flow with surgery on three-manifolds

Grisha Perelman¹

This is a technical paper, which is a continuation of [I]. Here we verify most of the assertions, made in [I, §13]; the exceptions are (1) the statement that a 3-manifold which collapses with local lower bound for sectional curvature is a graph manifold - this is deferred to a separate paper, as the proof has nothing to do with the Ricci flow, and (2) the claim about the lower bound for the volumes of the maximal horns and the smoothness of the solution from some time on, which turned out to be unjustified, and, on the other hand, irrelevant for the other conclusions.

The Ricci flow with surgery was considered by Hamilton [H 5,§ 4,5]; unfortunately, his argument, as written, contains an unjustified statement ( $R_{MAX}=\Gamma,$ on page 62, lines 7-10 from the bottom), which I was unable to fix. Our approach is somewhat different, and is aimed at eventually constructing a canonical Ricci flow, defined on a largest possible subset of space-time, - a goal, that has not been achieved yet in the present work. For this reason, we consider two scale bounds: the cutoff radius $h,$ which is the radius of the necks, where the surgeries are performed, and the much larger radius $r,$ such that the solution on the scales less than $r$ has standard geometry. The point is to make $h$ arbitrarily small while keeping $r$ bounded away from zero.