1547671870-The_Ricci_Flow__Chow

8 FINITE-TIME BLOWUP 209

and a sequence of times ti / T such that M (ti) ::; Ko. A consequence of
the differential inequality

:t 1Rm l^2 ::; 6. IRml^2 - 2 l\7 Rml^2 + Cn 1 Rml^3

we shall derive in Lemma 7.4 is the doubling-time estimate of Corollary 7.5,
which implies that there exists c > 0 depending only on the dimension n
such that

M(t)::; 2M(ti)::; 2Ko

for all times t satisfying

ti ::; t < min { T, ti + ;

0

}.

Since ti / T as i --+ oo, there exists io large enough that tio + c/ Ko 2: T.

But then this implies that

sup M (t) ::; 2Ko,

ti 0 9<T

which contradicts the claim established above and hence proves the theorem.

D

8. Finite-time blowup

In Section 9 below, we shall shift our attention to the normalized Ricci

fl.ow in order to complete our proof of Theorem 6.3. Along the way, we will

need to prove that the unique solution of the normalized Ricci fl.ow starting

on a closed 3-m anifold (M^3 , g 0 ) of positive Ricci curvature exists for all pos-

itive time. Paradoxically, the key to proving this fact is obtaining a better

understanding of exactly how the corresponding unnormalized solution be-

comes singular. We begin with the observation that a finite-time singularity

is inevitable.

LEMMA 6.53. Let (Mn, g (t) : 0 ::; t < T) be a solution of the unnormal-

ized Ricci .fiow for which the weak maximum principle holds. If there are

to 2: 0 and p > 0 such that

xEMinf n R (x, to) = p,

then g (t) becomes singular in .finite time.

PROOF. By equation (6.6), we have

:tR = 6.R + 2 IRcl

2

2: 6.R + ~R

2

.

Consider the solution

1

r (t) = -o-1- """""2 __ _

-p - Ti (t - to)