204 32. NONSINGULAR SOLUTIONS ON CLOSED 3-MANIFOLDS
3.2. Compactness theorem for the NRF.
The first main tool we shall use in the proof of Theorem 32.2 is the compactness
theorem. As in Definition 3.6 in Part I , we h ave
DEFINITION 32.4 (Pointed C^00 Cheeger- Gromov convergence). We say that asequence { (Mi, 9i ( t) , Xi)} iEl\I' t E (a, w), of pointed solutions to the NRF on closed
manifolds converges to a complete pointed family of metrics (M~, g 00 (t), x 00 ),
t E (a, w) , if there exist
(1) an exhaustion {Ui}iEl\I of Moo by open sets with Xoo E ui and
(2) a sequence of diffeomorphisms <I>i : Ui --+ Vi ~ <I>i (Ui) c Mi with
<I>i (xoo) =Xisuch that (Ui,i [gi(t)lvJ) converges in C^00 to (M 00 ,g 00 (t)) uniformly on com-
pact sets in M 00 x (a, w).
Let r 9 denote the average scalar curvature of g. The compactness theorem for
the NRF is as follows (this is completely analogous to Theorem 3.10 in Part I).
THEOREM 32.5 (Hamilton's Cheeger- Gromov-type compactness). Let{(Mi,gi (t) ,xi)}iEl\I' t E (a,w) 3 0, be a sequence of complete pointed solutions to
the NRF on closed manifolds such that
( 1) (uniformly bounded curvatures)1Rm 9 ,1 9 ,:::; Co on Mix (a,w)
for some constant Co < oo independent of i and
(2) ( injectivity radius estimate at t = 0)
inj 9 , (o) (xi) 2 lofor some constant lo > 0.
Then there exists a subsequence such that {(Mi, 9i (t), Xi)} iEl\I converges as i --+ oo
to a complete pointed solution (M~, 900 (t) , x 00 ) , t E (a,w), to the Ricci flow
with cosmological constant:
a 2
(32.5) ot (g 00 )iJ = -2 (Re 9 0JiJ + -;;_r 00 • (g 00 )iJ,where the .limit r 00 (t) ~ limHoo rg,(t) exists and the limit solution has uniformly
bounded curvature.
PROOF. First observe that rg,(t) is uniformly bounded because Rg,(t) is uni-
formly bounded. Applying the change of scale between Ricci fl.ow and NRF (see
(32.9)- (32.10) below) to Shi's derivative estimates, for any E > 0 we have bounds
on all derivatives of the curvature jvk Rm 9 , I :::; Ck on Mi x (a+ E, w), indepen-
dent of i. This implies, for each time derivative of the average scalar curvature, the
estimate I d:;r I :::; C~. For example, using (32.3), we compute that for a solution
g (t) to the NRF,
~: = :t (JM Rdμ I JM dμ)
= f~dμJM (21Rcl
2
-~rR+R(r-R))dμ.