262 9. TYPE I SINGULARITIES
in the sequel to this volume.) In terms of the solution g (t), this shows that
R (x, t) 2: Iv (x, t)I · [log ( Iv ~;lt)I) + log (1 + lvol t) - 3]
= Iv ( x, t) I. [log Iv ( x, t) I + log (I vo 1-^1 + t) - 3 J
wherever v (x, t) < 0. In particular, if v (x, t) < 0 at some point x E M^3
and time t > 0, then
R (x, t) > Jv (x, t)J (log Jv (x, t) I + logt - 3).Now if g (t) exists for -a ::; t < w, we can translate time to conclude that
R (x, t) > Jv (x, t) J (log Iv (x, t) J +log (t +a) - 3)
wherever v (x, t) < 0. But if g (t) is ancient, this leads to a contradiction,
since lima--+oo log (t +a) = oo. Hence v (x, t) 2: 0 for all x and t. 0
- Necklike points in finite-time singularities
Suppose (M^3 , g) is a closed solution of the Ricci fl.ow which forms a
singularity at T < oo, in other words, a singularity of Type I or Type
Ila. The main result of this section asserts that either M^3 is a spherical
space form or else M^3 admits a 'dimension reduction' somewhere near the
singularity, in the sense that at any time sufficiently close to T, we can find a
point in M^3 near which the geometry is arbitrarily close to the product of a
surface and a line. When the singularity is Type I, the surface arising from a
limit of dilations about a subsequence of such points and times approaching
T is a constant-curvature 2-sphere. In this case, the limit is a quotient of
S^2 x IR, and we say that M^3 develops a 'neck.' (See Section 5 Section of
Chapter 2.)
To make the notion of dimension reduction precise, we shall say that
(x, t) is a Type I c-essential point if
c
JRm (x, t) I 2: T _ t > o.
We say that (x, t) is a 8-necklike point if there exists a unit 2-form () at
(x, t) such that
JRm- R (0 ® O)J ::; 8 IRmJ.THEOREM 9.9. Let (M^3 , g (t)) be a closed solution of the unnormalized
Ricci flow on a maximal time interval 0 ::; t < T < oo. If the normalized
flow does not converge to a metric of constant positive sectional curvature,
then there exists a constant c > 0 such that for all T E [O, T) and 8 > 0,
there are x E M^3 and t E [T, T) such that (x, t) is a Type I c-essential point
and a 8-necklike point.Observe that the theorem is equivalent to the following statement, which
we will prove: