128 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS
for each s. We claim, for i sufficiently large, that
(29.210)
aw
as (s , ti)::; 0 for s::; Si·Indeed , if^88 1!' (s~, ti) > 0 for some s~ ::; Si, then^8 ; 8 ~
1> 0 implies that^88 1!" (si, ti ) > 0.
This in turn implies for s ;::=: Si that W(s, ti) 2". W(si, ti ) --+ oo, contradicting
(29.209) if i is sufficiently large.
Now, by (29.210) we have that
27rln(2AKf) ;::=: W(so+lnKi, t i ) 2". W(si,ti)--+ oo,
which contradicts K i --+ 0. This completes the proof of Claim l.
STEP 2. Pointed limits about (qi, ti)· Let i(s, e , t) ~ v(s +Si, e + ei, t +ti ),
where si and ei are as in (29.207) and ti --+ -oo is as before. Note that ?Ji ~
v;^1 (t)gcyl = e^28 '[!i(§(t +ti)), Where [Ii iS translation by (Si, ei) On the Cylinder
(on JR^2 this is multiplication by e^8 ' and rotation by ei)·
Define
W i(s, t) ~ r ln vi(s, e , t)dB = W(s +Si, t +ti)·
Js1
From (29.208), we have that 27rlnμ < Wi(O, O)::; C. By (29.206) we have
law as'(s,t) I <47r forsEJRandtE(-oo,-1-ti]·
Moreover, since t ~~ = R.'1 E (0, C],
!:l(sawi , t) = 1· R 9 (s + si , e + ei, t + ti )de E (O, 27rC]
ut S'
for s E JR and t E (-oo, - 1 - ti]· Thus, the Wi(s, t) are uniformly bounded oncompact subsets of JR x (-oo, oo) = JR^2. On the other hand, from (29.113) we
obtain I g 0 ln Vi I ::; C. Hence, the ln Vi are uniformly bounded on compact subsets
of JR x 51 x (-00,00). Since each Vi is a solution of (29.7), we obtain uniform
estimates for the higher derivatives of vi on compact subsets (see Lemma 28.53).
Therefore, there exists a subsequence such that the Vi converge in C^00 on compactsubsets to a smooth positive function v 00 on JR x 51 x ( -oo, oo) satisfying (29. 7).
That is, § 00 ~ v 00 (t)gcyl is a solution to the Ricci flow on JR x 51 x (-oo, oo). Note
that R9 00 (0, 0, 0) = limi->oo R9(si, Bi, ti)·
STEP 3. The pointed limit is a fiat cylinder.Claim 2. R9 00 (0, 0, 0) = 0.
Proof of Claim 2. Suppose that R 9 00 (0, 0, 0) > 0. We shall show that the
two cigar regions together with a neighborhood of (qi, ti) contribute integral scalar
curvature greater than 87f, yielding a contradiction.
We then have R 900 (o) > 0 on JR x 51. Hence there exists o > 0 such that
R 9 ,(o) 2:: o in [-1, 1] x 51 for i sufficiently large. Moreover , μ < vi(s, e , 0) ::; C in
[-1, 1] x 51. Therefore we have a given amount of total scalar curvature in a region
containing qi which is away from the "cigar region" near N:1
A-1 47fO. A. R 9 ,(o)vi (O)dsde ;:::: C =:= o.
[-l,ljxS^1