1547671870-The_Ricci_Flow__Chow

4. DILATIONS OF FINITE-TIME SINGULARITIES 241

PROOF. Recall from Lemma 7.4 that

:t 1Rml

2

::; ~ 1Rml

2

+ C 1Rml

3

for some C depending only on n. Define

K(t)~ sup 1Rm(x,t)l^2.

xEMn

By the maximum principle, we have dK / dt ::; C K^312 , which implies that

_!},__K-1/ 2 > -C_

dt - 2

Integrating this inequality from t to T E (t, T) and using the fact that

liminf K (T)-^1!^2 = 0,

T-+T

we obtain

K -^112 (t) ::; ~ (T - t).

Hence supxEMn jRm (x, t)I 2: 2/ [C (T-t)]. 0

4.1. Type I limits of Type I singularities. Given a Type I singular

solution (Mn,g(t)) on [O,T), define S by

(8.11) sup !Rm (x, t)i (T - t) ~ S < oo.

Mnx[O,T)

Let (xi, ti) be any sequence that is globally curvature essential. Then ti / T,
and the norm of the curvature at (xi, ti) is comparable to its maximum
at time tin the sense of (8.4). Consider the dilated solutions (Mn,gi(t))
of the Ricci fl.ow defined by (8.8), and recall that Rmi ~ Rm [gi] satisfies
IRmi (xi, O)I = 1 for all i.
For simplicity, we will assume the global bound (8.7); the general case
is essentially the same. By using definitions (8.8) and (8.11), estimate (8.4),
and Lemma 8.19, we find that the solutions (Mn, gi(t)) obey the uniform
bound

estimate

sup itl. IRmi(X, t)I ::; CS^2 < oo,

Mnx[-ai,O]