1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

294 7. THE REDUCED DISTANCE

where f3 : [O, 2v'TJ ~ Nn is a minimal constant speed geodesic with respect
to ho joining p to q. Thus

(7.24) L ( q, f) = d 2y'T (p, q)2

For reference below, we have L (q, f) ~ d (p, q)^2 and .e (q, f) ~ 2 ~L (q, f) =

d(~~)

2

; the definition of .e will be given again in (7.87).

SOLUTION TO EXERCISE 7.12. This exercise is a special case of the

discussion in Section 1 above. We also note that

(7.25)

I

r d d"( (r)^12 = I d d/3 (O')^12 = IVfg(O),^2

T g(T) (J' g(0'2/4)

where V ~ lim 7 -+o ft~~ (r) = lim0'-+0 ~~ (O'). We leave it to the reader to

check that for £-geodesics defined on a subinterval [ri, r 2 ] c [O, T], we still

have

(7.26) r I dd"f (r)l

2

=canst.

T g(T)

2.4. Elementary properties of L. In this subsection (Mn, g ( r)) , r E

[O, T], shall denote a complete solution to the backward Ricci flow, and

p E M shall be a basepoint. We will assume the curvature bound

(7.27) max {[Rm (x, r)f, /Re (x, r)f} :S: Co < oo.

(:z:,T)EMx[O,T]

The curvature bound assumption is written in this way for the convenience

of stating later estimates. We prove some elementary c^0 -estimates for the

£-distance and lengths of £-geodesics, relating them to the Riemannian

distance; we shall use these estimates often later.

First recall from (3.3) in Lemma 3.11 that for r1 < r2 and x EM,

e-2Co(T2-T1) g ( 72 , x) ::; g ( 71 , x) ::; e2Co(T2-T1) g ( 72 , x).

LEMMA 7.13 (£ and Riemannian distance). Let"( : [O, f] ~ M, f E

(0, T], be a C^1 -path starting at p and ending at q.

(i) (Bounding Riemannian distance by £) For any r E [O, f] we have

d;(O) (p, "Y ( r)) ::; 2fie2CoT (,e ( "!) + 2n3Co 73/2).

In particular 1 when M is noncompact, for any f E (0, T], we have

q-+oo lim L (q, f) = q-+oo lim 2vfiL (q, f) = +oo.

(ii) (Bounding speed at some time by £) There exists r* E (0, f) such

that

I

d"'f 1

2

I d/3 1

2

(^7) d (r) = d (O'*) ::; 2 1 '=,C ("Y) + -3-nCo 7 _ '

r g(T) O' g(T) yr