1547671870-The_Ricci_Flow__Chow

4. DILATIONS OF FINITE-TIME SINGULARITIES 245

or equivalently,

IR mi. ( x, t )I < ( ~ (~ - ) ti) I IRm ( (xi, ) ti)I I.

• c Ti - ti Rm Xi, ti - t
  Since Ti --) T, the Type Ila condition implies that

lim [ sup IRm (x, t)i (Ti - t)] = oo.

i-->oo Mn x [O,Ti]

Thus, by our choices of Xi and ti we have IRm (xi, ti)I (Ti - ti) --) oo.

Applying Theorem 8 .17 and the Compactness Theorem, one obtains a

limit solution (M~ 1 g 00 (t)) that exists for all t E (-00,00) with uniformly

bounded curvature

(8.18)

1

sup IRmool ~ -.

M~x(-00,00) C

As in the Type I case, a more carefully chosen sequence (xi, ti) will guar-

antee that the pointed limit solution (M~, g 00 (t), x 00 ) attains its maximum

curvature at ( x 00 , 0). Again choose any sequence of times Ti / T. The basic

idea is to choose points and times (xi, ti) such that

1

. IRm (xi, ti)I (Ti - ti)

(8.19) Im = 1.

i-->oo supMnx[O,T;] IRm (x, t)i (Ti - t)

We shall give the proof for the case that Mn is compact, leaving the general

case to the reader. Then since Ti < T, there exist points and times (xi, ti)

such that equality holds in (8.19). By repeating the argument of Section 4.2

with c = 1, we obtain the estimate

IR

mi. ( x, t )I ~ - (Ti --- ti) - IRm - -(xi---, ti)I

(Ti - ti) IRm (xi, ti)I - t

for all x E Mn and

t E [-ti IRm (xi, ti)I, (Ti - ti) IRm (xi, ti)I),

where we have

(Ti - ti) IRm (xi, ti)I--) oo.

Hence the pointed limit (M~,g 00 (t) ,x 00 ) is defined for all t E (-00,00)

and satisfies the uniform curvature bound

(8.20) sup 1Rmool 900 ~ 1.

M~x(-00,00)

Moreover, equality holds at (x 00 , 0), because IRmi (xi, O)I = 1.

REMARK 8.22. When n = 3, we have the estimate

CIRml- C' ~ R ~ C3 IRml

of Lemma 9.10, which implies that

sup R (x, t) (T - t) = oo.

M^3 x[O,T)