1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

7. THE £-FUNCTION ON EINSTEIN SOLUTIONS AND RICCI SOLITONS 337

Since by (7.103),

2fR(O) ~ R(O) -l

n R (r) '

we may rewrite (7.105) as

(7.106) R(q,f)='!!.(1-tan-l~(f)))+ d;(O)~,q) s(f) - ,

. 2 s (T) · 4T tan-^1 (s (T))'

where

s (f) ~ R R(f) (0) _ l.

Now we consider the extreme case: R (0) = oo.

EXERCISE 7.68. Let (Mn,g (T)) ,T > 0, be an Einstein solution of the

backward Ricci flow. Suppose that lim 7 ---+o R ( T) = oo so that R ( T) = 2 ~,

Re (T) = 2 ~g (T) and g (T) = Tg (1). Although the metric g (0) is not defined,

we may still consider (p, 0), p E M, as the basepoint for defining£, L and

Ras before. We have for/: [O, f]-+ M from p to q,

1

[, (r) = nVf + 7' T^3 /^2 ld'l2 d dT.

0 T g(l)

Show by considering the paths

;y (T) = { ~ (T/rJ)

where f3 : [O, 1] -+ M is a constant speed geodesic with respect to g (1)

joining p to q, and letting rJ-+ 0, that

(7.107) inf T^3 /^2 __:!_. dT = 0,

1

7' Id 12

I 0 dT g(l)

where the infimum is taken over / : [O, f] -+ M joining p to q. That is,

(7.108) L (q, f) = nv:f.

Hence for an Einstein solution g ( T) of the backward Ricci flow with

lim R (T) = oo,

T---+0

we have n

R(q,f) =

2

.

Using the rule for the R function on product spaces (see Exercise 7.11),

if we have a product solution (M x N, g (T) + h (T)), where (Mn, g (T)) is

an Einstein solution of the backward Ricci flow with lim 7 ---+0 R 9 ( T) = oo and
where (Nm, h ( T)) is Ricci fiat, then

9 +h ( _) n dh (p2, q2)

2

R(pi,p2,0) q1, q2, T = 2 + 4f