1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS 149

(2) By (7 .94) in Part I,

I~: I (x, t) ::; ~ JV'RJ2 + ~ JRI + 2 (~e~ t)'

Applying (30.60), JRI ::; ':.,

2

!'{, and (30.47) to this yields

I

ae I ( ) C^2 D

2

d~(t) (p, x) n^2 M

-xt<-- + +---

at ) -w - t (w - t)^2 2 (w - t)

Bd~(t) (p, x) ~ + 2n^2 M

+ 4 (w - t)^2 + 2 (w - t) ·

Now (30.61) follows easily. D

Recall that by (30.53), we have J! (x , t) + ~ :'.:'. 0 on M x [a+ c, w). We have

the following result of Naber [2 7 3] after Perelman.

LEMMA 30.17 (Lower bound for the reduced distance of a Type I solution).

Suppose that (Mn,g (t)), t E [a ,w], where -oo < a < w < oo, is a complete

solution of the Ricci flow with bounded curvature and satisfying the Type I condition

(30.45). Let p E M and let J! = Rp,w. Then for any c E (0, w2°') there exists a

constant C < oo depending only on n, M, and c such that for any x 1 , x 2 EM and

t E [a+ E,w) we have

(30.63)

V

A V A dg(t) (x1, x2)

J!(x1,t)+B+ J!(x2,t)+B2'. 1 1 - C,

4B2 (w - t)2

where A and B are given by (30.54) and (30.55). In particular, we have the fol-

lowing lower bound for the reduced distance:

(30.64). /J!(x,t) +A:'.:'. dgi(t) (x, p) i - C :'.:'. - C

V B 4B2 (w - t)^2

on M x [a + c , w).

PROOF. The idea is to follow the proof of Lemma 19.46 in Part III on bounding
the sum J! ( X1, t) + J! ( X2, t) of the reduced distance at two points at the same time,
while now using the Type I condition.
Given any x 1 , x 2 EM and any l E [a+ E,w), let ')' 1 : [t,w]---'* Mand ')'2 :

[t, w] ---'* M be minimal £-geodesics whose graphs join ( x 1 , l) and ( x 2 , l) to (p, w),

respectively. We have for t E [t, w],

(30.65) - :t ( dg(t) ('/'1 (t) '')'2 (t))) = - ( :t dg(t)) ('/'1 (t), ')'2 (t))

• t, \ V' adg(t) ('l'i(t), '/'2 (t)), dita (t)),

where the vector field V' adg(t) ( · , ·) denotes the gradient of the two-variable func-
tion dg(t) ( · , · ) with respect to the a-th variable.
Using (30.45), we can bound the term on the RHS of the first line of (30.65) by
applying Perelman's changing distances estimate. In particular, by (18.15)- (18.16)