1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

7. THE £-FUNCTION ON EINSTEIN SOLUTIONS AND RICCI SOLITONS 335

The second equality above needs uniform bound ..Ji I~:; 1;k (T) ::; C2, which

follows from Lemma 7.13(ii) and an argument similar to that in the proof of

Lemma 7.28. The last inequality above follows from the fact that <I>;;^1 o 'Yk

is a curve from p 00 to q. We have shown the desired opposite inequality

and the lemma is proved (we leave it as an exercise to prove the uniform

convergence). D

7. The .€-function on Einstein solutions and Ricci solitons

7.1. .e function on an Einstein solution with positive scalar cur-
vature. We consider an Einstein solution (Mn,g (T)) of the backward Ricci
fl.ow with positive scalar curvature defined on a time interval containing 0.
We first consider the case that the solution is smooth at T = 0 and then

generalize to the case that the solution becomes singular as T '\, 0. From

the scalar curvature evolution equation ~~ = -~ R^2 , we have

(7.103)

and

R ( T) - ------.,,^1 --

• R(0)-^1 + ~

l R(O)

1 + ~ R (0)

g(T)= (l+~R(o))g(O), TE (-2Rn(O)'oo).

Given a curve 'Y: [O, f]---+ M from p to q, we have

£ ( 'Y) = rr yT (R ( 'Y ( T) ' T) + I ~'Y 1

2

) dT

Jo T g(-r)

= rr yT ( !1 2 + (1 + 2T R (o)) I dd"j 12 ) dT.

Jo R (0) + ;; n T g(O)

Let a = 2../T, so that·

1

r ..Ji n

1

2...;¥ a2

--~--dT = - da

o R ( 0 )-^1 + ~ 2 o 2n · R ( 0 )-^1 + a^2

-n~ 1-.

( tan-

1

(Ju R ( 0) / n) )

J2TR(O) /n

Recall that the minimum of the functional J; <P ( T) x ( T)^2 dT under the

constraint J; x (T) dT =dis given by x ('r) = ¢(-r) J; <P(:)-1d 7. Similarly the

minimum of

(7.104)

1

r Id

1

E ('Y) ~ <P ( T) d 'Y^2 dT

0 T g(O)