1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

114 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS

(2) There exists a subsequence, still denoted by (qi, Ti), such that the

sequence (Mn,Ti-^1 9(TiT) ,qi) of dilated solutions of the backward

Ricci flow converges in the Cheeger-Gromov sense to a complete

nonfiat gradient shrinking Ricci soliton (M~,9oo (T) ,qoo)·

(3) The trace Harnack estimate holds for 900 ( T); hence the limit solu-

tion (M 00 ,9 00 (T)) is a f'i,-solution with Harnack.

(4) If n = 3, then (M~, 900 (T), q 00 ) has bounded Rm 00 2:: 0.

For example, the (unique up to homothety) asymptotic shrinker of either

(i) the Bryant soliton on JRn or (ii) Perelman's ancient solution on sn is the

shrinking cylinder sn-l x R

We now complete our presentation of the proof of Theorem 19.53. The

reader is assumed to have read §4 of Chapter 8 in Part I.

Given a /),-solution (Mn,9(8)), e E [O,oo), to the backward Ricci fl.ow

and given T > 0, define the rescaled solutions

9,,.(8) ~ T-l 9(TB),

for BE [O, oo).

STEP 1. Existence of a backward limit.

Choosing any sequence Ti ---+ oo and q 7 i E M so that

n

£ ( qTi , Ti) :S: 2

(such q 7 i exist by (7.15) of Perelman [152]; see also (7.95) in Part I), recall

that for a subsequence we have convergence in the C^00 pointed Cheeger-

Gromov sense

(19.60)

where 900 ( B) is a complete solution to the backward Ricci fl.ow which is f'i,-

noncollapsed on all scales, has Rmg 00 2:: O, and satisfies the trace Harnack

estimate.^29 In particular, there exist smooth embeddings

<l'>i : (Ui, qoo) C (Moo, qoo) ---+ (M, qTJ,

where {Ui} is an exhaustion of M 00 , such that

9i(e) ~ 1'>19TJe)---+ 9oo(B)

in each Ck norm, k EN, on compact subsets of M 00 •

STEP 2. Convergence of the reduced distances.

Fix a basepoint Po E M. We also have that the reduced distances of 97 i,

pulled back by <l'>i, converge to

(19. 61) .t:-i /) ::::;= • .c,(po,O) f)gTi o iii. 'l!'i ---+ .t:-oo, /}

where £ 00 is a locally Lipschitz function on M 00 x (0, oo) with \7 g 00 £ 00 and
:0£oo existing a.e. (a priori, it is possible that £ 00 #-gg= ).

(^29) In Theorem 8.32 of Part I, we stated this for () E ( A- (^1) , A) for arbitrary A > 1. The
convergence for () E (0, oo) follows from a standard diagonalization argument.