1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

332 7. THE REDUCED DISTANCE

since IY (7)1^2 = :¥ IY (r)l^2. Hence by (7.70),

(Hess(q,·r) L) (Y (r), Y (r))

:S~IY(r)l2 foT vr(1+T~7)Rd7+ IY,12

<: * ( 1 + T ~ f) IY (7)1

2

[VT ( R +I:~ I') dT + IY 'I'

= [ ( ~ + ~n!;) f (q,T) + ~ J IY(r)l

2

.

We have proved (compare with Lemma 7.41)

LEMMA 7.63. Let (Mn, g (7)), 7 E [O, T), be a solution to the backward

Ricci flow with bounded nonnegative curvature operator. We have for any

r E (0, T),

Recall that Hamilton's trace Harnack estimate says that

H ( X) ( 7) :::: - ( ~ + T ~

7

) R ( / ( 7) , 7).

Hence K (r, r) defined in (7.75) satisfies

K (r, r) :::: - fo:r 73 /^2 ( ~ + T ~

7

) R (r ( 7), 7) d7

'1T T ( Id 1

2

:::: - (^7) )
(^1) / (^2) ~ R (r ( 7) , 7) + di d7
0 T 7 7 g(T)

T ,

::::--T _L(r(r),r).

-7

Therefore, from (7.89); we have at (q, r)

2 1 T f

l\7RI :S -R + 73/2 T - TL (r (r) 'r) + f

:S -R + i (1 + T2~ r) f.

In particular,

LEMMA 7.64. Let (Mn,g(7)), 7 E [O,T), be a solution to the backward

Ricci flow with bounded nonnegative curvature operator. If r E (0, (1 - c) T)

for some c E (0, 1), then for any q EM