1547671870-The_Ricci_Flow__Chow

208 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

7.2. Long-time existence. Now we are ready to show that the cur-
vature becoming unbounded is the only obstacle to long-time existence of
the fl.ow.

PROOF OF THEOREM 6.45. Define M (t) ~ supxEMn IRm (x, t)i. We
shall first prove the claim that

limsup (M (t)) = oo.
t/T
By Theorem 3.13, a unique solution 9 (t) of the Ricci flow satisfying 9 (0) =
9o exists for a short time. Suppose that the solution exists on a maximal
finite time interval [O, T). To obtain a contradiction, suppose further that
there is a positive constant K such that

sup M (t) ::; K.

OSt<T

Fix a local coordinate patch U around an arbitrary point x E Mn, and let

T E (0, T) be arbitrary as well. Then by Lemma 6.49, a continuous limit

metric g (T) exists and is given in component form by the integral formula

9ij (x , T) = 9ij (x, T) - 21T Rij (x , t) dt.

Let a= (a1, ... ,ar) be any multi-index with lal = m EN. By Proposition
6 .48 and Corollary 6. 51, both g; 9ij and g; Rj are uniformly bounded on
U x [O, T). Thus we can write

(:;9ij) (x,T) = (:;9ij) (x,T)-21T (:;Rij) (x,t) dt,

which shows that la°'gl ::; C for some positive constant C, hence that g (T)

is a C^00 metric, and also

I ( :;9ij) (x, T) - ( :;9ij) (x , T)I ::; C (T - T),

which shows that g (T) ____, g (T) uniformly in any cm norm as t / T.

Now since 9 (T) is smooth, Theorem 3.13 implies there is a solution of

the Ricci flow g (t) that satisfies g (0) = g (T) and exists for a short time

0 ::; t < E. Since 9 ( T) ____, g (T) smoothly, if follows that

_. { g (t) 0::; t < T

g ( t)^7 g ( t - T) T ::; t < T + E

is a solution of the Ricci fl.ow that is smooth in space and time (in particular

near T) and satisfies 9 (0) = 90· This contradicts the assumption that T is

maximal, and proves the claim.

Finally, we prove the theorem by replacing the lim sup of the claim with

a bona fide limit. Suppose the theorem is false. Then there exists Ko < oo