1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

142 3. THE COMPACTNESS THEOREM FOR RICCI FLOW

on M 00 x (a,w) and Volg 00 (o)Bg 00 (o)(0 00 ,ro) 2:: vo. Furthermore Ooo is a

singular point in M 00 •

Idea of the proofs. Theorem 3.25 can be proved from Theorem

3.24 in the same way as we have proved Theorem 3.10 from Theorem

3.9. On the other hand, Theorem 3.24 can be proved with some modifi-

cation of the proof of Theorem 3.9 to handle the singularity (see [257]).

Note that the Bishop-Gromov volume comparison theorem holds for orb-

ifolds. Fix r > O; for any k and qk E Mk with dgk(o)(Ok,qk) Sr, we

have Volgk(O) Bgk(o)(qk,ro) 2:: v 1 , where v1 is a positive constant independent

of k but depending on v 0 , r, r 0 , n, Co. This implies that there exists rl in-

dependent of k such that Bgk(o) ( qk, rl) has the orbifold topological model

Bn /G (qk), where Bn is the unit ball in Euclidean space centered at the

origin and G (qk) C 0 (n) is a discrete subgroup with rank IG (qk)I bounded

independent of k. The existence of r 1 implies that we can modify the choice of

.\a in Definition 4.26 and Proposition 4.22 so that the ball Bk ~ B ( xk', .\a /2)

has the orbifold topological model Bn /G (xk). The key observation in the

proof of Theorem 3.24 is that we can choose a subsequence of orbifolds so

that the groups G (xk') and their actions on Bn are independent of k. We

can then use the balls Bk, Bk, Bk to build the limit orbifold.

4. Applications of Hamilton's compactness theorem

In this section we discuss some applications of Theorems 3.10 and 3.16.

We will see more applications of the compactness theorems later in this

volume.

4.1. Singularity models. Theorem 3.16 may be applied to study sin-

gular, nonsingular, and ancient solutions of the Ricci flow. For example,

let (Mn,g(t)), t E [O,T), where TE (O,oo], be a complete solution to

the Ricci fl.ow. Given a sequence of points and times {(xk, tk)}kEN, let

Kk ~ IRm (xb tk)I. We say that the sequence {(xk, tk)} satisfies an in-

jectivity :radius estimate if there exists lo > 0 independent of k such

that injg(tk) (xk) 2:: loK;,^112. Given a complete solution of the Ricci flow, we

can obtain a local limit of dilations provided we have an injectivity radius

estimate and a local bound on the curvatures after dilations.

COROLLARY 3.26 (Existence of singularity models). Let (Mn, g ( t)) , t E

(a, w), be a complete solution to the Ricci flow. Given a sequence of points

and times {(xk, tk)}kEN, let Kk ~ IRm (xk, tk)I > 0 and

9k (t) ~ Kkg (tk + K-;:^1 t).

Suppose that the sequence { ( Xk, tk)} satisfies an injectivity radius estimate,

i.e., injg(tk) (xk) 2:: loK"k^112 , for some lo > 0, and suppose that ak, Wk, a 00 , w 00

2:: 0 with ak ---+ a 00 > 0, Wk ---+ w 00 , and [tk - ~%, tk + 'k% J C (a, w) are such