1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. THE COMPACTNESS OF 11;-SOLUTIONS r43

ask--+ oo and for w E B-gk(o) (wk, sk),r^8

R-gk (w,O) ~ 2R-gk (wk,O).

Then applying the crucial Proposition 20.4 to the sequence of solutions

9k (t), t E (-oo, OJ, on B-gk(o) (wk, sk), we have that for every c > 0 there

exists A ( c) < oo independent of k such that

Vol-gk(o) B-gk(o)(wk, A (c) R-gk (wk, 0)-r/^2 )

(20.33) ~ c

(A (c) R-gk (wk, O)-r/^2 ) n

when k is large enough. Let co ~ ~ ( i1 6 ) n. From

R-gk (wk,O) · (3rk)^2 ;:::: R-gk (wk,O) · s~--+ oo,

we know that the radius in (20.33) with c = co satisfies

rk ~A (co) R-gk (wk, 0)-r/^2 ~ 4rk

fork sufficiently large.^19 Thus, by the Bishop-Gromov volume comparison
theorem (Rm;:::: 0 implies Re;:::: 0) and (20.33) we have

Vol-gk(O) Bgk(O) (Wk, 4rk) < Vol-gk(O) Bgk(O) (Wk, rk) < = '5:. (l)n

(4rkr - rk' - co 2 r6.

Since (20.32) holds for y =wk, i.e.,

Vol-gk(o) B-gk(O) (Wk, 4rk) ( r )n

(4rkr ::::: "' r6 ,

we obtain a contradiction. The claim is proved.

STEP 3. Uniform bound for R-gk (zk, 0).

Now that we have proved

(20.34)

where Cr < oo independent of k, we next show by a standard argument
using the trace Harnack estimate and Shi's local derivative estimate that
R-gk (zk, 0) has a uniform upper bound.
The integrated form of the trace Harnack estimate for the ancient solu-
tion 9k (t) with tr = -ar~ and t2 = 0 (see Exercise 15.6 in Part II) implies

(20.35)

R-gk (xk, 0)

R-gk (zk, -ar~)

(^18) In particular, for k large, the point Wk has large curvature and is the center of a
large (compared to the curvature at the center) ball where the curvatures are bounded
relative to the value at the center.
(^19) That is, the large scale A (c: 0 ), relative to the curvature at Wk, for the radius rk is
'embedded' in the fixed scale 4, relative to the curvature at Zk, for the radius 4rk defined
in (20.28).