64 28. SPECIAL ANCIENT SOLUTIONS
PROOF. STEP l. Diameter bound. Let g (T) = g (-T). By the Type l as-
sumption, we h ave Rcg(r)::; (n -1) ~^1 on M x [1,oo) for so me Ci. Suppose that
qi , q2 EM and f E [1, oo) are such that
d9(r) (qi, q2) > 2;-;;:.
Let (T., f], where T* E [1, f], be the maximal interval on which we h ave d9(r) (qi, q2)
2jf;; for TE (T., f]. Let/: [O,dg(r) (qi, q2)] ---+ M b e a unit sp eed minimal
geodes ic joining qi to q 2 with respect tog (T). By Proposition 18.8 in Part III, we
h ave
(28.71)for T E ( T*, f]. This implies
{
{!;_} 1
7
const (n, Ci)
dg(r) (qi , q2)::; max d9(i) (qi,q2) ,2y Ci + r. VT dT::; diam (g (1)) + const (n , Ci) ft.
Hence(28.72) diam (g (f))::; max { 2;-;;:, diam (g (1)) + const (n , Ci) ft}
for f E [1, oo).
STEP 2. A backwards limit is a shrinker. By Theo rem 30.31 in Chapter 30 ,there exist s a backwards limit (M".': 00 ,g_ 00 (t)), t E (-oo,O), which is a gradient
shrinker with nonnegative curvature operator. From (28.72) and scaling,
diam (g_ 00 (t)) ::; co nst (n , Ci) Jftj,so that M _ 00 is compact and hence diffeomorphic to M.
STEP 3. The so lution has constant sectional curvature. We h ave the following:
(1) Since g (t) has positive curvature operator (PCO), Mis a topological spher-
ical sp ace form (see Bohm and Wilking's Theo rem 11 .2 in Part II).
(2) Any shrinking Ricci soliton g (t) on M with nonnegative curvature operator
must h ave constant sectional curvature; in particular, g_ 00 (t) h as constant sectional
curvature. More sp ecifically, Theorem 11 .2 in Part II says t hat this is true underthe stronger assumption that g (t) has PCO. On the other h and, if g (t) does not
h ave PCO, then the combination of the strong maximum principle for systems (see
Theorem 12 .53 in P art II), Berger 's holonomy classification theorem (see Theorem
7.35 in [77]), and (1) lead to a contradiction.
(3) Let v (g) b e the v-invariant as defined in (6.50) of P art I. By v-invariant
monotonicity (see Lemma 6.35 in P art I) and by Theorem 11.2 in Part II, we have
that
sup v (g) = v (g),
g
where g has co nstant sectional curvature and the supremum is taken over all metrics
with PCO on M.
Thus, by v-monotonicity again and (2), we h ave