- ANCIENT SOLUTIONS WITH POSITIVE CURVATURE 63
LEMMA 28.54 (Pointwise subconvergence to a flat cylinder). The sequence ofsolutions ([2,oo) x S^1 ,u(s+si,B,t)gcyt), t E (-oo,O), subconverges in each Ck-
norm on compact sets to a complete static fiat solution g:X, = Uoo9cyl on IR x S^1 ,
whereUoo (s, e, t) = _lim u (s +Si, e, t).
i-tooBy the fact that limp-too R(p, t) = 0 for each t, we have that g:X, is flat. Hence
there exists a n isometry <p between g:X, and C9cyt , for some constant c > 0. Since <p
is an orientation-preserving conformal diffeomorphism of (IR^2 - {O}, 9cyt), it must
be given by multiplication by a nonzero complex number; this implies that il, 00 is a
constant.
Using Lemma 28.54, we now prove
LEMMA 28.55 (Vanishing of the second boundary term in the limit). At each
time t E ( - oo, 0) we have(28.70)where 8D.i is as in Lemma 28.45.
PROOF. From g = e-f+^2 s9cyl we have f = -lnu+2s. Since u(s+si,B,t)
converges as i ---+ oo in each Ck on compact sets to a constant a, on 8D.i we have
lv
(^2) JI < ~ 1v (^2) JI - ~ 1v (^2) ln ul ---+ o
g g - a 9cyl 9cyl - a 9cyl 9cyl
as i---+ 00. Because maxani IV 9cyl ln ul9cyl ---+ 0 as i ---+ oo, we h ave
Iv fl < 2a-
(^112) 1v JI = 2a- (^1)! (^2) l-v ln u + 2~ I < 6a- (^1)! (^2) on 8D.i
g g - 9cyl 9cyl 9cyl OS -
9cyl
for i large enough. Hence
Hm { v(JV fl^2 )du = 2 _lim { ( V^2 f, V f ® v) du= 0. D
i-too Jani i-too Jani
4. Ancient solutions with positive curvature
In Chapter 19 of Part III we discussed Perelman's nonround ancient K-solution
on sn for n 2: 3. His solution has positive curvature operator and is Type II. In
this section we discuss two results which classify positively curved ancient solutions
as shrinking spherical space forms under certain hypotheses.
4.1. Type I K-noncollapsed ancient solutions with positive curvature
operator (PCO).
To obtain more detailed information of Type I K-noncollapsed ancient solutions,
we make some extra assumptions. In this subsection we classify such solutions with
positive curvature operator. We have the following result proven by one of the
authors [285]. The rough idea is to use a diameter bound to apply the equality
case of Perelman's v-invariant.
THEOREM 28.56 (Compact Type I K-solutions with PCO). If (Mn, g (t)), t E
( -oo, 0), is a Type I K-noncollapsed (on all scales) ancient solution to the Ricci
flow with positive curvature operator and is defined on a compact man if old, then
(M, g (t)) is isometric to a shrinking spherical space form.