1547671870-The_Ricci_Flow__Chow

250 8. DILATIONS OF SINGULARITIES

fort > -a. Hence the pointed limit solution (M~, 900 (t), x 00 ), if it exists,

is defined for t E (-a, oo) and satisfies

a

sup IRmool :S --= IRmoo (xoo, O)I.

M~x(-n,oo) a+ t

6. Taking limits backwards in time

Assume that we have taken the limit of a sequence of dilations and

obtained a Type II ancient solution (M~,9 00 (t)) fort E (- oo,w), where

(8.27) sup lt l · IRmoo (x, t) I = oo.

M~x(-oo,OJ

Type II ancient limit solutions typically arise from taking a second limit,

as in dimension reduction. However, such limit solutions might not attain

their maximum curvatures, hence might not satisfy condition (8.1). When

n = 3, these limit solutions have nonnegative sectional curvature, which

allows for the application of the Harnack estimate. But to combine the

Harnack estimate with the strong maximum principle, it is necessary that

the limit solution attain its maximum scalar curvature. This motivates us

to take a another (third) limit to obtain a solution where this maximum

is attained. We can do this by translating and dilating about a suitable

sequence of times ti \,, -oo and points Xi E M~ where the curvatures

almost attain their maximums.

So let (M~, 900 (t)) be a complete Type II ancient solution defined for

t E ( - oo, w). Let ci < 1 be any sequence of constants such that ci \,, 0, and

choose any sequence of times Ti ---t -oo and corresponding points and times

(xi, ti) E M~ x (Ti, 0) such that

(8.28)

lti l (ti - Ti) IRmoo (xi, ti)I 2: (1 - ci) sup lt l (t - Ti) IRmoo (x, t)I.

M~xm,01

This is very similar to how we chose the points and times in the case of a

Type IIb singularity. Set

ai ~(ti - Ti) IRmoo (xi, ti)I 2: 0,

Wi ~-ti IRmoo (xi, ti)I 2: 0,

and observe that limi->oo ai = limi->oo Wi = oo. Indeed, by assumption

(8.27) and estimate (8.28), we have

1 aiWi ltil (ti - Ti) IRmoo (xi, ti)I

1/ai + l/wi ai + Wi ITil

2: (l _ ci) sup ltl (t - Ti) IRm^00 (x, t) I ---t 00

M~x[Ti,O] ITil

as i ---t oo, because Ti ..:....+ -oo.

Now consider the dilated solutions

(9oo)i (t) = IRmoo (xi, ti)I · 900 (ti+ IRmoot(xi, ti)I)