1547671870-The_Ricci_Flow__Chow

3. ANCIENT SOLUTIONS 33

In particular, the solution has positive curvature for as long as it exists.
Now regard M^2 as the 2-sphere 52 punctured at the poles:
IB.x5^1 ~ 52 \ { north and south poles }.

Since g (t) has positive scalar curvature for all t < 0, it is reasonable to
investigate under what conditions g (t) extends to a smooth solution of the

Ricci fl.ow on 52. We want to apply Lemma 2.10 with z = x and <p = 1./J =

y'U. Letting s (x, t) denote the distance from the 'equator' x = 0 to the

point x E IR at time t , we estimate

s (x, t) = J2,6 sinh (- a)d) dx

1

x 1

o v cosh ax + cosh a>..t

r lxl

~ 2J ,6 sinh (-a>..t) Jo e- %1xl dx.

Hence the distance to the 'poles' x = ±oo is bounded at all times t < 0

by ~ J ,6 sinh (-a>..t) < oo. This shows that (2.15) is satisfied. It is easy to
check that for all t < 0, we have

lim u ( x, t) = 0,

JxJ-->oo

hence that (2.16) is satisfied. Since

lim (fxJu(x,t)) =-a lim ( sinhax ) = =r=a

x-->±oo Ju ( x, t) x-->±oo cosh ax + cosh a>..t '

we see that (2.17) is satisfied if and only if a = 1. Finally, we compute

.§___Vu = 1 .!!.Vu = ~ .!!. log u = _ a sinh ax

as Vu ax 2 ax 2 (coshax + cosha>..t)

and recall (2.24) to see that

a

2

2

Vu = 1 a

2

2

log u = _ a

2

cosh a>..t · cosh ax + 1

2

.

as 2y'U ax 2J2,6 sinh (-a>..t) ( cosh ax+ cosh a>..t)^31

It follows that (2.18) is satisfied for k = 1; higher derivatives may be esti-

mated similarly. Applying Lemma 2.10, we get the following result.

LEMMA 2.12. The metric defined by (2.22) for t < 0 extends to an

ancient solution of the Ricci flow on 52 if and only if

1

(2.27) a = 1 and ,6 = 2 >...

From now on, we assume (2.27) holds. Then it follows from (2.26) that

the scalar curvature R±oo (t) at the 'poles' x = ±oo is strictly positive for

all times t < 0:

( )

.

1

. R
1
. >.. ( cosh >..t · cosh x + 1)
R±oo t :::;= JxJim -->oo = JxJ-->oo im sinh ( ->..t ) ( cosh x + cosh >..t )

= >..coth(->..t) > 0.