206 32. NONSINGULAR SOLUTIONS ON CLOSED 3 -MANIFOLDS3.4. Division of the study of nonsingular solutions into three cases.
We shall show that for any noncollapsed nonsingular solution to the NRF on
a closed 3-manifold, there exists such a limit which admits a metric with constant
curvature.
The properties of any limit (M~, g= (t), xcxo) in (32.8) depend on the asymp-
totic behavior of Rmin (t). Accordingly, we divide the analysis into three cases. As
we shall see, the more positive the curvature, the easier the analysis.
CASE I. (Positive) Rmin ( t 0 ) is positive for some to E [O, oo).CASE II. (Zero) limt-t= Rmin (t) = 0.
CASE III. (Negative) limt-tcxo Rmin (t) < 0.
In Cases I and II, it is useful to consider the corresponding solution to the Ricci
flow. Let (M^3 , g (i)) be the solution to gtgij = -2Rij with g (0) = g (0). The two
solutions g and g differ by the following change of scale in space and time:(32.9) g(t)='lf;(i)g(i),
where(32.10) 'lj; (i) =exp ( ~ ll f (i) di) =exp(~ lat r (T) dT).
Let Rmax (i) ~ maxxEM R9 (x, i).
LEMMA 32.8 (Criterion for g (t) to have nonnegative curvature). Let (M^3 , g (t)),
t E [O, oo), be a noncollapsed nonsingular solution to the NRF flow on a closed 3-
manifold. Let [O, T) be the maximal time interval for the corresponding solution
g (i) to the Ricci flow. Suppose that ti~ J:i 'lj; (T) dT, where {ii} satisfies(32.11) ii --+ T and ii'lj; (ii) --+ oo.
Then for any {xi} satisfying (32.6) and limit solution (M~, gcxo(t), x=), t E
(-00,00), to the NRF corresponding to {(xi, ti)}, we have that Mcxo is diffeomor-
phic to M and gcxo(t) is either fiat or has positive sectional curvature. Thus M is
diff eomorphic to a space form with nonnegative curvature.
PROOF. STEP 1. gcxo(t) has nonnegative sectional curvature. Let v 9 denote the
smallest eigenvalue of Rm9. We may assume that v 9 ( · , 0) is negative somewhere(otherwise we are done). Let C ~ - infxEM v 9 (x, 0) > 0. Then the Hamilton-Ivey
estimate says that at any point and time where v 9 < 0, we have
lv9I (ln lv9J + ln (c-^1 + i) - 3) :::; R 9.
Thus- Rgi(t) Rg(W) ( ~ -1 ~ I I)
1- 1
= I ~ I 2: ln 'lj;(ti + t)(C +ti+ t) v 9 i(t) -3.
Vg;(t) l/g(t;+t)
Now, by (32.11), we have limi-+= 'lf;(ii)(c-^1 +ii)= oo. By (32.10),lln 'lj;~~t) I=~ llti+t r (T) dTI :::; C ltl.
Hence, we also have limi-+cxo'lf;((+t)(c-^1 +t-;+t) = oo for all t E (-00, 00). This
implies that sect(g= (t)) 2: 0.