1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. THE £-LENGTH AND THE L-DISTANCE 295

where f3 (CT)~ I (r), CT= 2..,/T, and CT*~ 2~.

(iii) (Bounding L by Riemannian distance) For any q EM and 7 > 0,

L ( q,r -) < _ e 2Cor d;(r) (p, q) 2nCo -3/2

2

J7 + -

3

-r.

REMARK 7.14. In each of the estimates above, on the RHS one may think

of the first term as the main term and the second term as an error term.

Recall by (7.24) that if (Mn, g ( r) = go) is Ricci fl.at and I : [O, 7] ---+ M is a

minimal £-geodesic from p to q, then

d~ 0 (p, I (r)) = 2yfiL (r (r), r) = 2yfi..C ( li[o, 7 ]) = ~d~ 0 (p, q),

and for all r* E (0, 7) ,

I

d1 ( ) 12 - d;o (p, q)

r* r* -.

dr g(T*) 47

Hence for Ricci fl.at solutions, f (1 (r), r) defined in (7.87) is constant (=

4 ~d; 0 (p, q)) along £-geodesics.

PROOF. (i) Let 0-= 2.JT and f3 (0-) ~I (i). The idea is to first bound

the energy of /31 [o, 2 .J7']. By splitting the formula for ..C into two time intervals,

we see that

1

2.J7' I -d/3 (a-) I (^2) d0-
0 dO- g(a-2 I 4)
2../¥ 2 'F
=£(1)- r l~~(a-)1 da-- r v'f-R(1(i),i)di

12.J7' CT g(0-2/4) lo

(7.28) :::; ..C(1) + 2n3Co73/2,

since R ~ -nC 0. Hence, since g (0) :::; e^2007 g (i) for i E [O, r], we have

d;(o) (p, I ( r)) :::; e2CoT ( {2.J7' I~~ ( 0-) I - da-) 2

lo CT g(cr2/4)

:::; e2CoT. 2yfi {2.J7' I d~ ( 0-) 12 d0-

1 o dCT g(0-2/4)

:::; 2 .J!e2CoT (..c ( 1 ) + 2n

3

Co 7 3/2).

(ii) From the proof of (i) we have (taker= 7 in (7.28))

1 1

2

../¥ I d/3 1

2

• ---;:-(0-) dO-:::; -_..C^1 (1) + -7. nC^0
  2VT O dCT g(0-2/4) 2VT 3