1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

1. INTRODUCTION; STATEMENTS OF THE COMPACTNESS THEOREMS 131

(2) (injectivity radius estimate)

injgk (Ok) 2:: lo

for some constant lo > 0.

Then there exists a subsequence {jkhEN such that {(Mjk' 9jk, Ojk)}kEN

converges to a complete pointed Riemannian manifold (M~, 900 , 000 ) as

k-+ 00.

For sequences of solutions to the Ricci flow the corresponding conver-

gence theorem takes the following form.

THEOREM 3.10 (Compactness for solutions). Let {(Mk, 9k (t), Ok)}kEN,

t E (a, w) 3 0, be a sequence of complete pointed solutions to the Ricci flow

such that

(1) (uniformly bounded curvatures)

IRmklk :S Co on Mk x (a,w)

for some constant Co < oo independent of k and

(2) (injectivity radius estimate at t = 0)

injgk(o) (Ok) 2:: lo

for some constant lo > 0.

Then there exists a subsequence {jkhEN such that {(Mjk, 9jk (t), Ojk)}kEN

converges to a complete pointed solution to the Ricci flow ( M~, 900 ( t) , 0 00 ) ,

t E (a, w), as k -+ oo. -

Note that the second theorem only supposes bounds on the curvature,

not bounds on the derivatives of the curvature. This is because, for the

Ricci fl.ow, if the curvature is bounded on (a, w) , then all derivatives of the

curvature are bounded at times t > a (see Chapter 7 of Volume One or

Theorems A.29 and A.30 of this volume).^1 In particular all derivatives of

the curvature are bounded at time t = 0 and we can apply Theorem 3.9 to

. {(Mk,9k(O),Ok)}kEN'
In the next section we follow the proofs of Hamilton in [187]. We shall
assume Theorem 3.9, which will be proven in Chapter 4. We will show that

if there is a subsequence such that (Mk, 9k (0) , Ok) converges to a complete

limit (M~, 900 (0), 000 ), then there is a subsequence (Mk, 9k (t), Ok) which

converges at all times.

(^1) The bounds on the derivatives of Rm get worse as t --+ a.