1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

108 19. GEOMETRIC PROPERTIES OF 1;;-SOLUTIONS

PROOF. We follow the aforementioned references. Let '"Ya be a minimal

£-geodesic whose graph (ra (r) ,r) joins (po,O) to (qa,f) for a= 1,2. The

distance between q1 and q2 at time 7 may be expressed as

dg('F) (q1,q2) = 1

7

d~ (d 9 ( 7 ) (11 (r) ,/2 (r))) dr

(19.47) = 1

7

(:

7

d 9 ( 7 )) (ri(r) ,/2 (r)) dr

2 r7 I d )

+~Jo \V'adg( 7 )(ri(r),12(r)), dr;(r) dr,

where \7 adg(T) ( ·, ·) denotes the gradient of dg(T) ( ·, ·) with respect to the

a-th variable. Recall from Riemannian geometry

(19.48)

and by (7.54) in Part I (space-time Gauss lemma)

~r; (r) = \7/l, (ra (r), r).

Furthermore, the second inequality in Lemma 7.64 of Part I (see also
the original (7.16) in Perelman [152]) says that for an ancient solution with
bounded nonnegative curvature operator^27

(19.49) IV'el^2 (q, r) + R (q, r) :::; -e^3 (q, r).

T

Therefore

I

d'"'( dra I (r) = IV'el (ra (r), r) :S ( ~e (^3) (ra (r), r) ) 1/2
On the other hand, again using R ;:::: 0, we have

(19.50)

(recall that /a (7) = qa)· Thus, for a= 1, 2 and T E (0, 7],

Id /a T 1/2 · -

I

-1/4

dr (r):SJ373/4e (qa,r).

(^27) Note that for Euclidean space, IV£1 (^2) (q, T) = lq~:gl^2 = ~£ (q, T).