1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

154 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

n n £

Since u 00 ~ (47r)-2 (w 00 - t)-^2 e-= satisfies D~u 00 = 0 by (30.80), we have

by Lemma 6.8 in Part I (i.e., the original Proposition 9.1 in Perelman [312]) that

0 = D~v 00 = -2 (w 00 - t) IRc 9 = +V'^9 = \i'^9 =t' 00 - 2 (w~ _ t)9ool

2

Uoo·

We conclude that

1

Re 9 +V'^9 = \i'^9 =t' - g = 0

=^00 2 (w 00 - t)^00

on M 00 x [t 1 , t2]· Now Theorem 30.13 is proved. D

3. Type I solutions have shrinker singularity models

In this section, by applying the reduced volume monotonicity based at the
singular time and Perelman's pseudolocality theorem, we show that any Type I
singular solution must have a nonfl.at shrinking GRS as a singularity model. Of
course, this does not rule out the possibility of other types of singularity models
associated to a Type I solution.
Let (Mn,g (t)), t E [O,w), be a Type I singular solution on a closed manifold.
Given any Ai ---+ oo, we may rescale the solution by defining

(30.83)

fort E [->.iw, 0). This is equivalent to the "parabolic rescaling" corresponding to
the sequence of times ti ~ w - >-i^1 , which is defined by rescaling by the inverse of
the time to blow up:

gi ( t) ~ ( w - ti) -l g (ti + ( w - ti) t).

In particular, we have the correspondence

Note that since IRml (x, t) ::; w~t for some M < oo,

(30.84)

for t E [->.iw, 0). Hence, given any sequence Pi E M, by Perelman's no local
collapsing theorem and Hamilton's Cheeger- Gromov compactness theorem, there
exists a subsequence { (M, gi (t), Pi)} which converges to a complete a ncient solution

(30.85) (M~,g 00 (t) , p 00 ), t E (-oo, O),

with 1Rm 9 = I (x , t) ::; Jtl'; i.e., g 00 (t) is a Type I ancient solution. For this Cheeger-

Gromov convergence, let {Ui} be the corresponding exhaustion of M 00 and let

(30.86) i: (Ui,Poo)---+ (M, pi)