1547671870-The_Ricci_Flow__Chow

200 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

where C and I == 5 /2 depend only on go. On the other hand, it follows from
formula (6.6) that

.§_R^2 -^1 = ~ (R^2 -^1 ) - (2 -1) (1 -1) R-^1 l\7 Rl^2 + 2 (2 -1) R^1 -^1 IRcl^2.

8t

Choose "/3 depending only on go such that

and recall that

{3-< (Rmin (0))

1

O< - 3(2- 1)(1-1)'

IV Rl^2 :S 3 l\7 Rcl^2.

(This is a consequence of (6.46), but is easy to prove directly.) Then for any

f3 E [O, "/JJ, one has an estimate

:t (V - {3R2-1) :S ~ (V - /3R2-1)

• [!3 (2 - 1) (1 - 1) R-^1 1v R l^2 - IV Rcl^2 ]

+ CR^3 -^21 - 2(3 (2 -1) R^1 -^1 IRcl^2

:S ~ (V - {3R^2 -^1 ) + C1,

where C1 depends only on f3 and go. By the maximum principle, we conclude

that

V - {3R^2 -^1 :S Cit.

But formula (6.6) implies that

.§_R 8t > - ~R + ~R 3 2 · )

as we shall see in Lemma 6.53, this forces the solution to become singular

within a finite time T depending only on go. Hence we conclude that

IV R l2 < v < {3R2-1 + c T.

R - - i

D

7. Higher derivative estimates and long-time existence

If go is a smooth metric on a compact manifold Mn, Theorem 3.13

implies that a unique solution g (t) of the Ricci fl.ow satisfying g (0) = go

exists for a short time. It follows that there is a maximal time interval

0 :S t < T :S oo on which the solution exists. If T < oo, it will be important

for us to understand what goes wrong. The goal of this section is to prove

the following result.