1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

38 28. SPECIAL ANCIENT SOLUTIONS

STEP 5. ODI comparison gives a lower bound for Smin· Recall that in general

if we have a solution to the ODE

(28.16) ds =A (s^2 - B^2 )

dt

on [o:, to] withs (to) =so, where A , B > 0 and so < 0, then

(28.17)

Ce-2BA(to-t) + 1

s (t) = -B Ce-2BA(to-t) - 1'

where C = ~~~~ provided B "I--so. If B =-so, thens (t) =so.

Take

(28.18)

2

A=~ -c:,

B = _1_ const

JA2V€p2'

and s 0 = Smin (to) < 0. It follows from (28.15) and the ODI comparison theorem

that we have Smin (t) ::; s (t) and s (t) > -oo fort E [o:, to] (the latter inequality is

since Smin (t) > -oo).

CASE 1. If s 0 2 - B , then we have the estimate

(28.19) R(O, to) 2 Smin (to) 2 -B.

CASE 2. If so < -B, t hen C > 1 and since s (t) > -oo fort E [o:, to], we h ave

that the denominator of S ( o:) satisfies

so that

So - B e-2BA(t 0 -a) _ l > O

s 0 +B '

e2BA(to-a) + 1

(28.20) R ( 0 , to) 2 so > -B e 2 BA(to-a) _

1

·

Since (28.19) is a stronger estimate t han (28.20), we conclude that (28.20) holds

in either case for any p 2 r 0 + ~(n - l)r 01 (to - o:). Note that, from (28.18), A

depends only on n and c:, whereas B depends only on n, c:, and p. Further note

that although r 0 is used to define i' and hence S, in the end, we shall obtain a lower

estimate for R independent of r 0.

STEP 6. Sharp lower bound for R. Now fix c: E (0 , ~) and let p---+ oo. We then

h ave B ---+ 0. Since by l'Hopital's rule

e2BA(to-a) + (^1 1)
B--to limB------e2BA(to-a) _ 1
A (to - o:)
and since (28.20) holds for all p , we conclude that
1
R ( 0, to) 2 - ( 2 ) (.
n -c: to - o:)
Finally, taking c;---+ 0, we obtain
n
R ( 0 , to) 2 - 2 ( t )
0 - o:

Since our only assumption on ( 0 , to) E M x ( o:, w) was R ( 0 , t 0 ) < 0, this estimate

holds everywh ere. D

By taking o:---+ -oo in Theorem 28.1, we obtain the following, which does not

assume a curvature bound.